<reponame>UCD4IDS/sage<gh_stars>0
r"""
Relative Number Fields

AUTHORS:

- <NAME> (2004, 2005): initial version
- <NAME> (2006-05-12): added support for relative extensions
- <NAME> (2007-09-04): major rewrite and documentation
- <NAME> (2008-10): specified embeddings into ambient fields
- <NAME> (2009-01): modernize coercion implementation
- <NAME> (2012-08): added is_CM_extension
- <NAME> (2014-04): absolute number fields are unique parents

This example follows one in the Magma reference manual::

    sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000)
    sage: z = y^5/11; z
    420/11*y^3 - 40000/11*y
    sage: R.<y> = PolynomialRing(K)
    sage: f = y^2 + y + 1
    sage: L.<a> = K.extension(f); L
    Number Field in a with defining polynomial y^2 + y + 1 over its base field
    sage: KL.<b> = NumberField([x^4 - 420*x^2 + 40000, x^2 + x + 1]); KL
    Number Field in b0 with defining polynomial x^4 - 420*x^2 + 40000 over its base field

We do some arithmetic in a tower of relative number fields::

    sage: K.<cuberoot2> = NumberField(x^3 - 2)
    sage: L.<cuberoot3> = K.extension(x^3 - 3)
    sage: S.<sqrt2> = L.extension(x^2 - 2)
    sage: S
    Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
    sage: sqrt2 * cuberoot3
    cuberoot3*sqrt2
    sage: (sqrt2 + cuberoot3)^5
    (20*cuberoot3^2 + 15*cuberoot3 + 4)*sqrt2 + 3*cuberoot3^2 + 20*cuberoot3 + 60
    sage: cuberoot2 + cuberoot3
    cuberoot3 + cuberoot2
    sage: cuberoot2 + cuberoot3 + sqrt2
    sqrt2 + cuberoot3 + cuberoot2
    sage: (cuberoot2 + cuberoot3 + sqrt2)^2
    (2*cuberoot3 + 2*cuberoot2)*sqrt2 + cuberoot3^2 + 2*cuberoot2*cuberoot3 + cuberoot2^2 + 2
    sage: cuberoot2 + sqrt2
    sqrt2 + cuberoot2
    sage: a = S(cuberoot2); a
    cuberoot2
    sage: a.parent()
    Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field

WARNING: Doing arithmetic in towers of relative fields that depends on
canonical coercions is currently VERY SLOW.  It is much better to
explicitly coerce all elements into a common field, then do arithmetic
with them there (which is quite fast).

TESTS::

    sage: y = polygen(QQ,'y'); K.<beta> = NumberField([y^3 - 3, y^2 - 2])
    sage: K(y^10)
    27*beta0
    sage: beta^10
    27*beta0
"""
# ****************************************************************************
#       Copyright (C) 2004-2009 <NAME> <<EMAIL>>
#                     2014-2022 <NAME> <<EMAIL>>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  https://www.gnu.org/licenses/
# ****************************************************************************

import sage.libs.ntl.all as ntl

from sage.categories.map import is_Map
from sage.structure.sequence import Sequence

import sage.structure.parent_gens

from . import maps
from . import structure

from sage.misc.latex import latex
from sage.misc.cachefunc import cached_method
from sage.structure.factorization import Factorization
import sage.rings.polynomial.polynomial_element as polynomial_element
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

from . import number_field_element
import sage.rings.number_field.number_field_ideal_rel
from .number_field_ideal import is_NumberFieldIdeal
from .number_field import (NumberField, NumberField_generic,
    put_natural_embedding_first, proof_flag,
    is_NumberFieldHomsetCodomain)
from sage.rings.number_field.number_field_base import is_NumberField
from sage.rings.number_field.order import (RelativeOrder, is_NumberFieldOrder,
                                           relative_order_from_ring_generators)
from sage.rings.number_field.morphism import RelativeNumberFieldHomomorphism_from_abs
from sage.libs.pari.all import pari_gen

from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ


def is_RelativeNumberField(x):
    r"""
    Return ``True`` if `x` is a relative number field.

    EXAMPLES::

        sage: from sage.rings.number_field.number_field_rel import is_RelativeNumberField
        sage: is_RelativeNumberField(NumberField(x^2+1,'a'))
        False
        sage: k.<a> = NumberField(x^3 - 2)
        sage: l.<b> = k.extension(x^3 - 3); l
        Number Field in b with defining polynomial x^3 - 3 over its base field
        sage: is_RelativeNumberField(l)
        True
        sage: is_RelativeNumberField(QQ)
        False
    """
    return isinstance(x, NumberField_relative)


class NumberField_relative(NumberField_generic):
    """
    INPUT:

    - ``base`` -- the base field

    - ``polynomial`` -- a polynomial which must be defined in the ring `K[x]`,
      where `K` is the base field.

    - ``name`` -- a string, the variable name

    - ``latex_name`` -- a string or ``None`` (default: ``None``), variable name
      for latex printing

    - ``check`` -- a boolean (default: ``True``), whether to check
      irreducibility of ``polynomial``

    - ``embedding`` -- currently not supported, must be ``None``

    - ``structure`` -- an instance of :class:`structure.NumberFieldStructure`
      or ``None`` (default: ``None``), provides additional information about
      this number field, e.g., the absolute number field from which it was
      created

    EXAMPLES::

        sage: K.<a> = NumberField(x^3 - 2)
        sage: t = polygen(K)
        sage: L.<b> = K.extension(t^2+t+a); L
        Number Field in b with defining polynomial x^2 + x + a over its base field

    TESTS::

        sage: Z = var('Z')
        sage: K.<w> = NumberField(Z^3 + Z + 1)
        sage: L.<z> = K.extension(Z^3 + 2)
        sage: K = loads(dumps(L))
        sage: K
        Number Field in z with defining polynomial Z^3 + 2 over its base field
        sage: L is K
        True

        sage: M.<u,v> = L.change_names()
        sage: M.structure()
        (Isomorphism given by variable name change map:
          From: Number Field in u with defining polynomial x^3 + 2 over its base field
          To:   Number Field in z with defining polynomial Z^3 + 2 over its base field,
         Isomorphism given by variable name change map:
          From: Number Field in z with defining polynomial Z^3 + 2 over its base field
          To:   Number Field in u with defining polynomial x^3 + 2 over its base field)
        sage: loads(dumps(M)) is M
        True
    """
    def __init__(self, base, polynomial, name,
                 latex_name=None, names=None, check=True, embedding=None, structure=None):
        r"""
        Initialization.

        EXAMPLES::

            sage: K.<x> = CyclotomicField(5)[]
            sage: W.<a> = NumberField(x^2 + 1)
            sage: W
            Number Field in a with defining polynomial x^2 + 1 over its base field
            sage: type(W)
            <class 'sage.rings.number_field.number_field_rel.NumberField_relative_with_category'>

        Test that check=False really skips the test::

            sage: W.<a> = NumberField(K.cyclotomic_polynomial(5), check=False)
            sage: W
            Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1 over its base field

        A relative extension of a relative extension::

            sage: x = polygen(ZZ)
            sage: k.<a0,a1> = NumberField([x^2 + 2, x^2 + 1])
            sage: l.<b> = k.extension(x^2 + 3)
            sage: l
            Number Field in b with defining polynomial x^2 + 3 over its base field
            sage: l.base_field()
            Number Field in a0 with defining polynomial x^2 + 2 over its base field
            sage: l.base_field().base_field()
            Number Field in a1 with defining polynomial x^2 + 1

        Non-monic and non-integral polynomials are supported (:trac:`252`)::

            sage: l.<b> = k.extension(5*x^2 + 3); l
            Number Field in b with defining polynomial 5*x^2 + 3 over its base field
            sage: l.pari_rnf()
            [x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4), ..., y^4 + 6*y^2 + 1, x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4)], [0, 0]]
            sage: b
            b

            sage: l.<b> = k.extension(x^2 + 3/5); l
            Number Field in b with defining polynomial x^2 + 3/5 over its base field
            sage: l.pari_rnf()
            [x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4), ..., y^4 + 6*y^2 + 1, x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4)], [0, 0]]
            sage: b
            b

            sage: l.<b> = k.extension(x - 1/a0); l
            Number Field in b with defining polynomial x + 1/2*a0 over its base field
            sage: l.pari_rnf()
            [x, [4, -x^3 - x^2 - 7*x - 3, -x^3 + x^2 - 7*x + 3, 2*x^3 + 10*x], ..., [x^4 + 6*x^2 + 1, -x, -1, y^4 + 6*y^2 + 1, x], [0, 0]]
            sage: b
            -1/2*a0

        TESTS:

        Test that irreducibility testing is working::

            sage: x = polygen(ZZ)
            sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 3])
            sage: K.<a> = NumberField(x^2 + 2)
            sage: x = polygen(K)
            sage: L.<b> = K.extension(x^3 + 3*a)

            sage: (x^3 + 2*a).factor()
            (x - a) * (x^2 + a*x - 2)
            sage: L.<b> = K.extension(x^3 + 2*a)
            Traceback (most recent call last):
            ...
            ValueError: defining polynomial (x^3 + 2*a) must be irreducible
            sage: (x^2 + 2).factor()
            (x - a) * (x + a)
            sage: L.<b> = K.extension(x^2 + 2)
            Traceback (most recent call last):
            ...
            ValueError: defining polynomial (x^2 + 2) must be irreducible
            sage: L.<b> = K.extension(x^2 + 2)
            Traceback (most recent call last):
            ...
            ValueError: defining polynomial (x^2 + 2) must be irreducible

        Error checks::

            sage: x = polygen(ZZ)
            sage: K.<a> = NumberField(x^2 + 1)
            sage: K.extension(x^2 + 2, 'a')
            Traceback (most recent call last):
            ...
            ValueError: base field and extension cannot have the same name 'a'
        """
        if embedding is not None:
            raise NotImplementedError("Embeddings not implemented for relative number fields")
        if names is not None:
            name = names
        if not is_NumberField(base):
            raise TypeError("base (=%s) must be a number field"%base)
        if not isinstance(polynomial, polynomial_element.Polynomial):
            try:
                polynomial = polynomial.polynomial(base)
            except (AttributeError, TypeError):
                raise TypeError("polynomial (=%r) must be a polynomial" % polynomial)
        if name == base.variable_name():
            raise ValueError("base field and extension cannot have the same name %r" % name)
        if polynomial.parent().base_ring() != base:
            polynomial = polynomial.change_ring(base)
            #raise ValueError, "The polynomial must be defined over the base field"

        # Generate the nf and bnf corresponding to the base field
        # defined as polynomials in y, e.g. for rnfisfree

        # Convert the polynomial defining the base field into a
        # polynomial in y to satisfy PARI's ordering requirements.

        if base.is_relative():
            abs_base = base.absolute_field(name + '0')
            from_abs_base, to_abs_base = abs_base.structure()
        else:
            abs_base = base
            from_abs_base = maps.IdentityMap(base)
            to_abs_base = maps.IdentityMap(base)

        self.__absolute_base_field = abs_base, from_abs_base, to_abs_base
        self.__base_field = base
        self.__relative_polynomial = polynomial
        self._element_class = number_field_element.NumberFieldElement_relative

        if check and not self.pari_relative_polynomial().polisirreducible():
            raise ValueError("defining polynomial (%s) must be irreducible"%polynomial)

        names = (name,) + base.variable_names()
        self._assign_names(tuple(names), normalize=False)

        NumberField_generic.__init__(self, self.absolute_polynomial(), name=None,
                                     latex_name=latex_name, check=False,
                                     embedding=embedding, structure=structure)

        self._zero_element = self(0)
        self._one_element =  self(1)

    def change_names(self, names):
        r"""
        Return relative number field isomorphic to self but with the
        given generator names.

        INPUT:

        - ``names`` -- number of names should be at most the number of
          generators of self, i.e., the number of steps in the tower
          of relative fields.

        Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where
        from_K is an isomorphism from `K` to self and ``to_K`` is an
        isomorphism from self to `K`.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: L.<c,d> = K.change_names()
            sage: L
            Number Field in c with defining polynomial x^4 + 3 over its base field
            sage: L.base_field()
            Number Field in d with defining polynomial x^2 + 2

        An example with a 3-level tower::

            sage: K.<a,b,c> = NumberField([x^2 + 17, x^2 + x + 1, x^3 - 2]); K
            Number Field in a with defining polynomial x^2 + 17 over its base field
            sage: L.<m,n,r> = K.change_names()
            sage: L
            Number Field in m with defining polynomial x^2 + 17 over its base field
            sage: L.base_field()
            Number Field in n with defining polynomial x^2 + x + 1 over its base field
            sage: L.base_field().base_field()
            Number Field in r with defining polynomial x^3 - 2

        And a more complicated example::

            sage: PQ.<X> = QQ[]
            sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
            sage: PF.<Y> = F[]
            sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
            sage: L.<m, n, r> = K.change_names(); L
            Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field
            sage: L.structure()
            (Isomorphism given by variable name change map:
              From: Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field,
             Isomorphism given by variable name change map:
              From: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              To:   Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field)
        """
        if len(names) == 0:
            names = self.variable_names()
        elif isinstance(names, str):
            names = names.split(',')
        K = self.base_field().change_names(tuple(names[1:]))
        to_K = K.structure()[1]
        old_poly = self.relative_polynomial()
        new_poly = PolynomialRing(K, 'x')([to_K(c) for c in old_poly])
        return K.extension(new_poly, names=names[0], structure=structure.NameChange(self))

    def subfields(self, degree=0, name=None):
        """
        Return all subfields of this relative number field self of the given degree,
        or of all possible degrees if degree is 0.  The subfields are returned as
        absolute fields together with an embedding into self.  For the case of the
        field itself, the reverse isomorphism is also provided.

        EXAMPLES::

            sage: PQ.<X> = QQ[]
            sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
            sage: PF.<Y> = F[]
            sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
            sage: K.subfields(2)
            [
            (Number Field in c0 with defining polynomial x^2 - 24*x + 96, Ring morphism:
              From: Number Field in c0 with defining polynomial x^2 - 24*x + 96
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c0 |--> -4*b + 12, None),
            (Number Field in c1 with defining polynomial x^2 - 24*x + 120, Ring morphism:
              From: Number Field in c1 with defining polynomial x^2 - 24*x + 120
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c1 |--> 2*b*a + 12, None),
            (Number Field in c2 with defining polynomial x^2 - 24*x + 72, Ring morphism:
              From: Number Field in c2 with defining polynomial x^2 - 24*x + 72
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c2 |--> -6*a + 12, None)
            ]
            sage: K.subfields(8, 'w')
            [
            (Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9, Ring morphism:
              From: Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: w0 |--> (-1/2*b*a + 1/2*b + 1/2)*c, Relative number field morphism:
              From: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              To:   Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9
              Defn: c |--> -1/3*w0^7 + 4*w0^5 - 12*w0^3 + 11*w0
                    a |--> 1/3*w0^6 - 10/3*w0^4 + 5*w0^2
                    b |--> -2/3*w0^6 + 7*w0^4 - 14*w0^2 + 6)
            ]
            sage: K.subfields(3)
            []
        """
        if name is None:
            name = self.variable_name()
        abs = self.absolute_field(name)
        from_abs, to_abs = abs.structure()
        abs_subfields = abs.subfields(degree=degree)
        ans = []
        for K, from_K, to_K in abs_subfields:
            from_K = K.hom([from_abs(from_K(K.gen()))])
            if to_K is not None:
                to_K = RelativeNumberFieldHomomorphism_from_abs(self.Hom(K), to_K*to_abs)
            ans.append((K, from_K, to_K))
        ans = Sequence(ans, immutable=True, cr=ans!=[])
        return ans

    def is_absolute(self):
        r"""
        Returns False, since this is not an absolute field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.is_absolute()
            False
            sage: K.is_relative()
            True
        """
        return False

    def gens(self):
        """
        Return the generators of this relative number field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.gens()
            (a, b)

        TESTS:

        Trivial extensions work like non-trivial ones (:trac:`2220`)::

            sage: NumberField([x^2 - 3, x], 'a').gens()
            (a0, 0)
            sage: NumberField([x, x^2 - 3], 'a').gens()
            (0, a1)

        """
        return ((self._gen_relative(),) +
                tuple(map(self, self.base_field().gens())))

    def _first_ngens(self, n):
        """
        Return the first `n` generators of this relative number field.

        If `n` is greater than the number of generators, the output is
        the same as that of :meth:`gens`.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K._first_ngens(0)
            ()
            sage: K._first_ngens(1)
            (a,)
            sage: K._first_ngens(2)
            (a, b)
            sage: K._first_ngens(3)
            (a, b)
        """
        if n <= 0:
            return ()
        v = (self._gen_relative(),)
        if n > 1:
            v += tuple(map(self, self.base_field()._first_ngens(n - 1)))
        return v

    def ngens(self):
        """
        Return the number of generators of this relative number field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.gens()
            (a, b)
            sage: K.ngens()
            2
        """
        return self.base_field().ngens() + 1

    def gen(self, n=0):
        """
        Return the `n`'th generator of this relative number field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.gens()
            (a, b)
            sage: K.gen(0)
            a
        """
        if n == 0:
            return self._gen_relative()
        return self(self.base_field().gen(n - 1))

    def galois_closure(self, names=None):
        r"""
        Return the absolute number field `K` that is the Galois closure of this
        relative number field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.galois_closure('c')
            Number Field in c with defining polynomial x^16 + 16*x^14 + 28*x^12 + 784*x^10 + 19846*x^8 - 595280*x^6 + 2744476*x^4 + 3212848*x^2 + 29953729
        """
        return self.absolute_field('a').galois_closure(names=names)

    def composite_fields(self, other, names=None, both_maps=False, preserve_embedding=True):
        """
        List of all possible composite number fields formed from self and
        other, together with (optionally) embeddings into the compositum;
        see the documentation for both_maps below.

        Since relative fields do not have ambient embeddings,
        preserve_embedding has no effect.  In every case all possible
        composite number fields are returned.

        INPUT:

        - ``other`` - a number field

        - ``names`` - generator name for composite fields

        - ``both_maps`` - (default: False)  if True, return quadruples
          (F, self_into_F, other_into_F, k) such that self_into_F maps self into
          F, other_into_F maps other into F.  For relative number fields k is
          always None.
        - ``preserve_embedding`` - (default: True) has no effect, but is kept
          for compatibility with the absolute version of this function.  In every
          case the list of all possible compositums is returned.

        OUTPUT:

        -  ``list`` - list of the composite fields, possibly with maps.


        EXAMPLES::

            sage: K.<a, b> = NumberField([x^2 + 5, x^2 - 2])
            sage: L.<c, d> = NumberField([x^2 + 5, x^2 - 3])
            sage: K.composite_fields(L, 'e')
            [Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600]
            sage: K.composite_fields(L, 'e', both_maps=True)
            [[Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600,
              Relative number field morphism:
              From: Number Field in a with defining polynomial x^2 + 5 over its base field
             To:   Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600
              Defn: a |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e
                    b |--> -21/166400*e^7 + 73/20800*e^5 - 779/10400*e^3 + 7/260*e,
              Relative number field morphism:
              From: Number Field in c with defining polynomial x^2 + 5 over its base field
              To:   Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600
              Defn: c |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e
                    d |--> -3/25600*e^7 + 7/1600*e^5 - 147/1600*e^3 + 1/40*e,
              None]]
        """
        if not isinstance(other, NumberField_generic):
            raise TypeError("other must be a number field.")
        if names is None:
            sv = self.variable_name()
            ov = other.variable_name()
            names = sv + (ov if ov != sv else "")

        self_abs = self.absolute_field('w')
        abs_composites = self_abs.composite_fields(other, names=names, both_maps=both_maps)

        m = self.absolute_degree()

        if not both_maps:
            rets = []
            for F in abs_composites:
                if F.absolute_degree() == m:
                    F = self
                rets.append(F)
            return rets

        from_self_abs, to_self_abs = self_abs.structure()

        rets = []
        for F, self_abs_to_F, other_to_F, k in abs_composites:
            self_to_F = RelativeNumberFieldHomomorphism_from_abs(self.Hom(F), self_abs_to_F*to_self_abs)
            if F.absolute_degree() == m:
                if other.is_absolute():
                    other_to_F = other.hom([(from_self_abs*(~self_abs_to_F)*other_to_F)(other.gen())])
                else:
                    other_to_F = RelativeNumberFieldHomomorphism_from_abs(self.Hom(self), from_self_abs*(~self_abs_to_F)*other_to_F)
                self_to_F = RelativeNumberFieldHomomorphism_from_abs(self.Hom(self), from_self_abs)
                F = self
            rets.append([F, self_to_F, other_to_F, None])
        return rets

    def absolute_degree(self):
        """
        The degree of this relative number field over the rational field.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.absolute_degree()
            6
        """
        return self.absolute_polynomial().degree()

    def relative_degree(self):
        r"""
        Returns the relative degree of this relative number field.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.relative_degree()
            2
        """
        return self.relative_polynomial().degree()

    def degree(self):
        """
        The degree, unqualified, of a relative number field is deliberately
        not implemented, so that a user cannot mistake the absolute degree
        for the relative degree, or vice versa.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.degree()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field you must use relative_degree or absolute_degree as appropriate
        """
        raise NotImplementedError("For a relative number field you must use relative_degree or absolute_degree as appropriate")

    @cached_method
    def _maximal_order(self, v=(), assume_maximal='non-maximal-non-unique'):
        """
        Implements :meth:`NumberField_generic.maximal_order` for relative
        number fields.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.maximal_order() is K.maximal_order()  # indirect doctest
            True

        """
        absolute_order = self.absolute_field('z').maximal_order(v=v, assume_maximal=assume_maximal)

        return RelativeOrder(self, absolute_order, is_maximal=assume_maximal, is_maximal_at=v)

    def _repr_(self):
        """
        Return string representation of this relative number field.

        The base field is not part of the string representation.  To
        find out what the base field is use :meth:`~base_field`.

        EXAMPLES::

            sage: k.<a, b> = NumberField([x^5 + 2, x^7 + 3])
            sage: repr(k) # indirect doctest
            'Number Field in a with defining polynomial x^5 + 2 over its base field'
            sage: k.base_field()
            Number Field in b with defining polynomial x^7 + 3
        """

        return "Number Field in %s with defining polynomial %s over its base field"%(self.variable_name(), self.relative_polynomial())

    def _Hom_(self, codomain, category=None):
        """
        Return homset of homomorphisms from this relative number field
        to the codomain.

        EXAMPLES:

        This function is implicitly called by the ``Hom`` method or
        function::

            sage: K.<a,b> = NumberField([x^3 - 2, x^2+1])
            sage: K.Hom(K) # indirect doctest
            Automorphism group of Number Field in a with defining polynomial x^3 - 2 over its base field
            sage: type(K.Hom(K))
            <class 'sage.rings.number_field.homset.RelativeNumberFieldHomset_with_category'>

        TESTS::

            sage: H = End(K)
            sage: loads(dumps(H)) is H
            True
        """
        if not is_NumberFieldHomsetCodomain(codomain):
            raise TypeError("{} is not suitable as codomain for homomorphisms from {}".format(codomain, self))

        from sage.rings.number_field.homset import RelativeNumberFieldHomset
        return RelativeNumberFieldHomset(self, codomain, category)

    def _latex_(self):
        r"""
        Return a `\LaTeX` representation of the extension.

        EXAMPLES::

            sage: x = polygen(QQ)
            sage: K.<a> = NumberField(x^3 - 2)
            sage: t = polygen(K)
            sage: K.extension(t^2+t+a, 'b')._latex_()
            '( \\Bold{Q}[a]/(a^{3} - 2) )[b]/(b^{2} + b + a)'
        """
        latex_name = self.latex_variable_names()[0]
        return "( %s )[%s]/(%s)"%(latex(self.base_field()), latex_name,
                              self.relative_polynomial()._latex_(latex_name))

    def _coerce_from_other_number_field(self, x):
        """
        Coerce a number field element x into this number field.

        In most cases this currently doesn't work (since it is
        barely implemented) -- it only works for constants.

        INPUT:

        - ``x`` -- an element of some number field

        EXAMPLES::

            sage: K.<a> = NumberField(x^3 + 2)
            sage: L.<b> = NumberField(x^2 + 1)
            sage: K._coerce_from_other_number_field(L(2/3))
            2/3
        """
        if x.parent() is self.base_ring():
            return self.__base_inclusion(x)

        f = x.polynomial()
        if f.degree() <= 0:
            return self._element_class(self, f[0])
        # todo: more general coercion if embedding have been asserted
        raise TypeError("Cannot coerce element into this number field")

    def _convert_non_number_field_element(self, x):
        r"""
        Convert the non-number field element `x` into this number field.

        INPUT:

        - ``x`` -- a non number field element, e.g., a list,
          integer, rational, or polynomial.

        EXAMPLES::

            sage: x = polygen(ZZ)
            sage: K.<a> = NumberField(x^5 + 2)
            sage: L.<b> = K.extension(x^2 + 3*a)
            sage: R.<u> = QQ[]
            sage: S.<t> = R[]

            sage: L(a + b)
            b + a

            sage: L(5*t*(1 + u) + 2/3*u)
            (5*a + 5)*b + 2/3*a
            sage: L(0*t + 2/3)
            2/3
            sage: L(1/2*t + 5)
            1/2*b + 5

        This seems reasonable::

            sage: L(t*5)
            5*b

        This is misleading, but correct!  It is more often desired
        to make a number field element given by rational
        coefficients of the relative power basis (so 2*b^2 + 3)
        than it is to create the constant term of such an element,
        which is what would happen if L(u*5) gave 5*a.::

            sage: L(u*5)
            5*b

            sage: L([1, 1/2])
            1/2*b + 1
            sage: L([ a, 1/2 + a/3 ])
            (1/3*a + 1/2)*b + a

            sage: L([ 1 ])
            Traceback (most recent call last):
            ...
            ValueError: Length must be equal to the degree of this number field

        TESTS:

        Examples from :trac:`4727`::

            sage: K.<j,b> = QQ[sqrt(-1), sqrt(2)]
            sage: j
            I
            sage: j.list()
            [0, 1]
            sage: K(j.list())
            I
            sage: (b*j + 1/2).list()
            [1/2, sqrt2]
            sage: K((b*j + 1/2).list())
            sqrt2*I + 1/2

        Examples from :trac:`4869`::

            sage: K.<z> = CyclotomicField(7)
            sage: Ky.<y> = PolynomialRing(K)
            sage: L.<a> = K.extension(y^2 + 1)
            sage: K(K.polynomial_ring().random_element()) # random
            -12*z^2 + 1/2*z - 1/95
            sage: L(L.polynomial_ring().random_element()) # random
            (z^5 + 1/3*z^4 - z^3 + z^2 - z + 2/3)*a + 1/4*z^5 - 7/2*z^4 + 5/3*z^3 - 1/4*z^2 + 3/2*z - 1

        Examples from :trac:`11307`::

            sage: L = NumberField([x^2 + 1, x^2 - 3], 'a')
            sage: L(L)
            Traceback (most recent call last):
            ...
            TypeError: unable to convert Number Field in a0 with defining polynomial x^2 + 1 over its base field to Number Field in a0 with defining polynomial x^2 + 1 over its base field
            sage: L in L
            False

        We construct the composite of three quadratic fields, then
        coerce from the quartic subfield of the relative extension::

            sage: k.<a,b,c> = NumberField([x^2 + 5, x^2 + 3, x^2 + 1])
            sage: m = k.base_field(); m
            Number Field in b with defining polynomial x^2 + 3 over its base field
            sage: k(m.0)
            b
            sage: k(2/3)
            2/3
            sage: k(m.0^4)
            9

            sage: x = polygen(ZZ)
            sage: K.<a> = NumberField(x^2 + 2, 'a')
            sage: L.<b> = K.extension(x - a, 'b')
            sage: L(a)
            a
            sage: L(b+a)
            2*a
            sage: K.<a> = NumberField(x^5 + 2, 'a')
            sage: L.<b> = K.extension(x - a, 'b')
            sage: L(a)
            a
            sage: L(a**3)
            a^3
            sage: L(a**2+b)
            a^2 + a
            sage: L.<b> = K.extension(x + a/2, 'b')
            sage: L(a)
            a
            sage: L(a).polynomial()
            -x
            sage: L(a).minpoly()
            x - a
            sage: L(a).absolute_minpoly()
            x^5 + 2
            sage: L(b)
            -1/2*a
            sage: L(b).polynomial()
            1/2*x
            sage: L(b).absolute_minpoly()
            x^5 - 1/16
            sage: L(b).minpoly()
            x + 1/2*a

        ::

            sage: K.<a> = NumberField(x^5+2)
            sage: R.<y> = K[]
            sage: L.<x0> = K.extension(y + a**2)
            sage: L(a)
            a
        """
        if isinstance(x, polynomial_element.Polynomial):
            # we have been given a polynomial, change it to an absolute polynomial
            K = self.base_ring()
            R = self.polynomial_ring()
            if QQ.has_coerce_map_from(x.parent().base_ring()):
                # special case absolute polynomials -- they should be
                # in terms of the relative generator
                x = R(x.list())
            # this should work for base_ring()['x'] and QQ['base']['ext']
            x = self.polynomial_ring()(x)
            f = R( [ K(coeff) for coeff in x.list() ] )
            return self._element_class(self, f(self.gen()).polynomial() )

        # Anything else: use the code for generic number fields
        return super(NumberField_relative, self)._convert_non_number_field_element(x)

    def _coerce_map_from_(self, R):
        """
        Canonical coercion of x into this relative number field.

        Currently integers, rationals, the base field, and this field
        itself coerce canonically into this field (and hence so does
        anything that coerces into one of these).

        EXAMPLES::

            sage: k.<a> = NumberField([x^5 + 2, x^7 + 3])
            sage: b = k(k.base_field().gen())
            sage: b = k.coerce(k.base_field().gen()) # indirect doctest
            sage: b^7
            -3
            sage: k.coerce(2/3)
            2/3
            sage: c = a + b # no output
        """
        if R is int:
            return self._generic_coerce_map(R)
        elif R in (ZZ, QQ, self.base_field()):
            return self._generic_coerce_map(R)
        if is_NumberFieldOrder(R) and R.number_field() is self:
            return self._generic_coerce_map(R)
        mor = self.base_field()._internal_coerce_map_from(R)
        if mor is not None:
            return self._internal_coerce_map_from(self.base_field()) * mor

    def _element_constructor_(self, x, check=True):
        """
        Construct a relative number field element from ``x``.

        EXAMPLES:

        We can create relative number field elements from PARI::

            sage: y = polygen(QQ)
            sage: K.<a> = NumberField(y^2 + y + 1)
            sage: x = polygen(K)
            sage: L.<b> = NumberField(x^4 + a*x + 2)
            sage: e = a.__pari__(); e
            Mod(y, y^2 + y + 1)
            sage: L(e)  # Conversion from PARI base field element
            a
            sage: e = (a*b).__pari__('x'); e
            Mod(-x^4 - 2, x^8 - x^5 + 4*x^4 + x^2 - 2*x + 4)
            sage: L(e)  # Conversion from PARI absolute number field element
            a*b
            sage: e = L.pari_rnf().rnfeltabstorel(e); e
            Mod(Mod(y, y^2 + y + 1)*x, x^4 + Mod(y, y^2 + y + 1)*x + 2)
            sage: L(e)  # Conversion from PARI relative number field element
            a*b
            sage: e = pari('Mod(0, x^8 + 1)'); L(e)  # Wrong modulus
            Traceback (most recent call last):
            ...
            TypeError: cannot convert PARI element Mod(0, x^8 + 1) into Number Field in b with defining polynomial x^4 + a*x + 2 over its base field

        We test a relative number field element created "by hand"::

            sage: e = pari("Mod(Mod(y, y^2 + y + 1)*x^2 + Mod(1, y^2 + y + 1), x^4 + y*x + 2)")
            sage: L(e)
            a*b^2 + 1

        A wrong modulus yields an error::

            sage: e = pari('Mod(y*x, x^4 + y^2*x + 2)'); L(e)
            Traceback (most recent call last):
            ...
            TypeError: cannot convert PARI element Mod(y*x, x^4 + y^2*x + 2) into Number Field in b with defining polynomial x^4 + a*x + 2 over its base field
        """
        # If x is a *relative* PARI number field element, convert it
        # to an absolute element.
        if isinstance(x, pari_gen) and x.type() == "t_POLMOD":
            modulus = x.mod()
            if (modulus == self.pari_relative_polynomial()
                or modulus == self.pari_absolute_base_polynomial()):
                x = self._pari_rnfeq()._eltreltoabs(x.liftpol())
                check = False
        return NumberField_generic._element_constructor_(self, x, check=check)

    def __base_inclusion(self, element):
        """
        Given an element of the base field, give its inclusion into
        this extension in terms of the generator of this field.

        This is called by the canonical coercion map on elements from
        the base field.

        EXAMPLES::

            sage: k.<a> = NumberField([x^2 + 3, x^2 + 1])
            sage: m = k.base_field(); m
            Number Field in a1 with defining polynomial x^2 + 1
            sage: k.coerce(m.0 + 2/3) # indirect doctest
            a1 + 2/3
            sage: s = k.coerce(m.0); s
            a1
            sage: s^2
            -1

        This implicitly tests this coercion map::

            sage: K.<a> = NumberField([x^2 + p for p in [5,3,2]])
            sage: K.coerce(K.base_field().0)
            a1
            sage: K.coerce(K.base_field().0)^2
            -3

        TESTS:

        Check that :trac:`5828` is solved::

            sage: K.<w> = QuadraticField(-1)
            sage: KX.<X> = K[]
            sage: H.<h> = K.extension(X-1)
            sage: H(w)
            w
        """
        abs_base, from_abs_base, to_abs_base = self.absolute_base_field()
        # Write element in terms of the absolute base field
        element = self.base_field().coerce(element)
        element = to_abs_base(element)
        # Express element as a polynomial in the absolute generator of self
        nfzk = self._pari_nfzk()
        expr_x = self._pari_base_nf()._nfeltup(element._pari_polynomial(), nfzk)
        # We do NOT call self(...) because this code is called by
        # __init__ before we initialize self.gens(), and self(...)
        # uses self.gens()
        return self._element_constructor_(expr_x, check=False)

    def _fractional_ideal_class_(self):
        """
        Return the Python class used to represent ideals of a relative
        number field.

        EXAMPLES::

            sage: k.<a> = NumberField([x^5 + 2, x^7 + 3])
            sage: k._fractional_ideal_class_ ()
            <class 'sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel'>
        """
        return sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel

    def _pari_base_bnf(self, proof=False, units=True):
        r"""
        Return the PARI bnf (big number field) representation of the
        absolute base field in terms of the pari variable ``y``, suitable
        for extension by the pari variable ``x``.

        All caching is done by the absolute base field.

        INPUT:

        - ``proof`` (bool, default True) -- if True, certify
          correctness of calculations (not assuming GRH).

        EXAMPLES::

            sage: k.<a> = NumberField([x^3 + 2, x^2 + 2])
            sage: k._pari_base_bnf()
            [[;], matrix(0,3), [;], ...]
        """
        abs_base, from_abs_base, to_abs_base = self.absolute_base_field()
        return abs_base.pari_bnf(proof, units)

    def _pari_base_nf(self):
        r"""
        Return the PARI number field representation of the absolute
        base field, in terms of the pari variable ``y``, suitable for
        extension by the pari variable ``x``.

        All caching is done by the absolute base field.

        EXAMPLES::

            sage: y = polygen(QQ,'y')
            sage: k.<a> = NumberField([y^3 + 2, y^2 + 2])
            sage: k._pari_base_nf()
            [y^2 + 2, [0, 1], -8, 1, ..., [1, 0, 0, -2; 0, 1, 1, 0]]
        """
        abs_base, from_abs_base, to_abs_base = self.absolute_base_field()
        return abs_base.pari_nf()

    def is_galois(self):
        r"""
        For a relative number field, ``is_galois()`` is deliberately not
        implemented, since it is not clear whether this would mean "Galois over
        `\QQ`" or "Galois over the given base field". Use either ``is_galois_absolute()`` or ``is_galois_relative()`` respectively.

        EXAMPLES::

            sage: k.<a> =NumberField([x^3 - 2, x^2 + x + 1])
            sage: k.is_galois()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field L you must use either L.is_galois_relative() or L.is_galois_absolute() as appropriate
        """
        raise NotImplementedError("For a relative number field L you must use either L.is_galois_relative() or L.is_galois_absolute() as appropriate")

    def is_galois_relative(self):
        r"""
        Return True if for this relative extension `L/K`, `L` is a
        Galois extension of `K`.

        EXAMPLES::

            sage: K.<a> = NumberField(x^3 - 2)
            sage: y = polygen(K)
            sage: L.<b> = K.extension(y^2 - a)
            sage: L.is_galois_relative()
            True
            sage: M.<c> = K.extension(y^3 - a)
            sage: M.is_galois_relative()
            False

        The next example previously gave a wrong result; see :trac:`9390`::

            sage: F.<a, b> = NumberField([x^2 - 2, x^2 - 3])
            sage: F.is_galois_relative()
            True
        """
        d = self.relative_degree()
        if d <= 2:
            return True
        else:
            rel_poly = self.relative_polynomial()
            return d == len(rel_poly.base_extend(self).factor())

    def is_galois_absolute(self):
        r"""
        Return True if for this relative extension `L/K`, `L` is a Galois extension of `\QQ`.

        EXAMPLES::

            sage: K.<a> = NumberField(x^3 - 2)
            sage: y = polygen(K); L.<b> = K.extension(y^2 - a)
            sage: L.is_galois_absolute()
            False

        """
        f = self.absolute_polynomial()
        return f.galois_group(pari_group=True).order() == self.absolute_degree()

    def is_isomorphic_relative(self, other, base_isom=None):
        r"""
        For this relative extension `L/K` and another relative extension `M/K`, return True
        if there is a `K`-linear isomorphism from `L` to `M`. More generally, ``other`` can be a
        relative extension `M/K^\prime` with ``base_isom`` an isomorphism from `K` to
        `K^\prime`.

        EXAMPLES::

            sage: K.<z9> = NumberField(x^6 + x^3 + 1)
            sage: R.<z> = PolynomialRing(K)
            sage: m1 = 3*z9^4 - 4*z9^3 - 4*z9^2 + 3*z9 - 8
            sage: L1 = K.extension(z^2 - m1, 'b1')
            sage: G = K.galois_group(); gamma = G.gen()
            sage: m2 = (gamma^2)(m1)
            sage: L2 = K.extension(z^2 - m2, 'b2')
            sage: L1.is_isomorphic_relative(L2)
            False
            sage: L1.is_isomorphic(L2)
            True
            sage: L3 = K.extension(z^4 - m1, 'b3')
            sage: L1.is_isomorphic_relative(L3)
            False

        If we have two extensions over different, but isomorphic, bases, we can compare them by
        letting ``base_isom`` be an isomorphism from self's base field to other's base field::

            sage: Kcyc.<zeta9> = CyclotomicField(9)
            sage: Rcyc.<zcyc> = PolynomialRing(Kcyc)
            sage: phi1 = K.hom([zeta9])
            sage: m1cyc = phi1(m1)
            sage: L1cyc = Kcyc.extension(zcyc^2 - m1cyc, 'b1cyc')
            sage: L1.is_isomorphic_relative(L1cyc, base_isom=phi1)
            True
            sage: L2.is_isomorphic_relative(L1cyc, base_isom=phi1)
            False
            sage: phi2 = K.hom([phi1((gamma^(-2))(z9))])
            sage: L1.is_isomorphic_relative(L1cyc, base_isom=phi2)
            False
            sage: L2.is_isomorphic_relative(L1cyc, base_isom=phi2)
            True

        Omitting ``base_isom`` raises a ValueError when the base fields are not identical::

            sage: L1.is_isomorphic_relative(L1cyc)
            Traceback (most recent call last):
            ...
            ValueError: other does not have the same base field as self, so an isomorphism from self's base_field to other's base_field must be provided using the base_isom parameter.

        The parameter ``base_isom`` can also be used to check if the relative extensions are
        Galois conjugate::

            sage: for g in G:
            ....:   if L1.is_isomorphic_relative(L2, g.as_hom()):
            ....:       print(g.as_hom())
            Ring endomorphism of Number Field in z9 with defining polynomial x^6 + x^3 + 1
              Defn: z9 |--> z9^4
        """
        if is_RelativeNumberField(other):
            s_base_field = self.base_field()
            o_base_field = other.base_field()
            if base_isom is None:
                if s_base_field is o_base_field:
                    return self.relative_degree() == other.relative_degree() and len(self.relative_polynomial().roots(other)) > 0
                raise ValueError("other does not have the same base field as self, so an isomorphism from self's base_field to other's base_field must be provided using the base_isom parameter.")
            if s_base_field.absolute_degree() != o_base_field.absolute_degree():
                raise ValueError("The base fields are not isomorphic.")
            if base_isom.domain() is s_base_field and base_isom.codomain() is o_base_field:
                if s_base_field.absolute_degree() != o_base_field.absolute_degree():
                    raise ValueError("The base fields are not isomorphic.")
                if not self.relative_degree() == other.relative_degree():
                    return False
                R = PolynomialRing(o_base_field, 'x')
                F = R([base_isom(_) for _ in self.relative_polynomial()])
                return len(F.roots(other)) > 0
            raise ValueError("base_isom is not a homomorphism from self's base_field to other's base_field")
        raise ValueError("other must be a relative number field.")

    def is_CM_extension(self):
        """
        Return True is this is a CM extension, i.e. a totally imaginary
        quadratic extension of a totally real field.

        EXAMPLES::

            sage: F.<a> = NumberField(x^2 - 5)
            sage: K.<z> = F.extension(x^2 + 7)
            sage: K.is_CM_extension()
            True
            sage: K = CyclotomicField(7)
            sage: K_rel = K.relativize(K.gen() + K.gen()^(-1), 'z')
            sage: K_rel.is_CM_extension()
            True
            sage: F = CyclotomicField(3)
            sage: K.<z> = F.extension(x^3 - 2)
            sage: K.is_CM_extension()
            False

        A CM field K such that K/F is not a CM extension

        ::

            sage: F.<a> = NumberField(x^2 + 1)
            sage: K.<z> = F.extension(x^2 - 3)
            sage: K.is_CM_extension()
            False
            sage: K.is_CM()
            True

        """

        try:
            return self.__is_CM_extension
        except(AttributeError):
            pass

        if self.relative_degree() == 2:
            if self.base_field().is_totally_real():
                if self.is_totally_imaginary():
                    self.__is_CM_extension = True
                    self.__is_CM = True
                    self.__max_tot_real_sub = [self.base_field(), self._internal_coerce_map_from(self.base_field())]
                    return True
        self.__is_CM_extension = False
        return False

    @cached_method(key=lambda self, base, basis, map: (base or self.base_ring(), basis, map))
    def free_module(self, base=None, basis=None, map=True):
        """
        Return a vector space over a specified subfield that is isomorphic to this number field,
        together with the isomorphisms in each direction.

        INPUT:

        - ``base`` -- a subfield

        - ``basis`` -- (optional) a list of elements giving a basis over the subfield

        - ``map`` -- (default ``True``) whether to return isomorphisms to and from the vector space

        EXAMPLES::

            sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K
            Number Field in a with defining polynomial x^2 + 2 over its base field
            sage: V, from_V, to_V = K.free_module()
            sage: to_V(K.0)
            (0, 1)
            sage: W, from_W, to_W = K.free_module(base=QQ)
            sage: w = to_W(K.0); len(w)
            18
            sage: w[0]
            -127917622658689792301282/48787705559800061938765
        """
        if basis is not None:
            raise NotImplementedError
        if base is None:
            base = self.base_field()
        if base is self.base_field():
            V = self.base_field()**self.relative_degree()
            if not map:
                return V
            fr = maps.MapRelativeVectorSpaceToRelativeNumberField(V, self)
            to = maps.MapRelativeNumberFieldToRelativeVectorSpace(self, V)
        elif base is QQ:
            if not map:
                return QQ**self.absolute_degree()
            K = self.absolute_field('a')
            from_K, to_K = K.structure()
            V, from_V, to_V = K.free_module()
            fr = maps.MapVectorSpaceToRelativeNumberField(V, self, from_V, from_K)
            to = maps.MapRelativeNumberFieldToVectorSpace(self, V, to_K, to_V)
        else:
            raise NotImplementedError
        return V, fr, to

    def relative_vector_space(self, base=None, *args, **kwds):
        """
        Return vector space over the base field of self and isomorphisms
        from the vector space to self and in the other direction.

        EXAMPLES::

            sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K
            Number Field in a with defining polynomial x^2 + 2 over its base field
            sage: V, from_V, to_V = K.relative_vector_space()
            sage: from_V(V.0)
            1
            sage: to_V(K.0)
            (0, 1)
            sage: from_V(to_V(K.0))
            a
            sage: to_V(from_V(V.0))
            (1, 0)
            sage: to_V(from_V(V.1))
            (0, 1)

        The underlying vector space and maps is cached::

            sage: W, from_V, to_V = K.relative_vector_space()
            sage: V is W
            True
        """
        if base is not None and base is not self.base_field():
            raise ValueError("Relative vector space base must be the base field")
        return self.free_module(self.base_field(), *args, **kwds)

    def absolute_vector_space(self, base=None, *args, **kwds):
        r"""
        Return vector space over `\QQ` of ``self`` and isomorphisms from
        the vector space to ``self`` and in the other direction.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^3 + 3, x^3 + 2]); K
            Number Field in a with defining polynomial x^3 + 3 over its base field
            sage: V,from_V,to_V = K.absolute_vector_space(); V
            Vector space of dimension 9 over Rational Field
            sage: from_V
            Isomorphism map:
              From: Vector space of dimension 9 over Rational Field
              To:   Number Field in a with defining polynomial x^3 + 3 over its base field
            sage: to_V
            Isomorphism map:
              From: Number Field in a with defining polynomial x^3 + 3 over its base field
              To:   Vector space of dimension 9 over Rational Field
            sage: c = (a+1)^5; c
            7*a^2 - 10*a - 29
            sage: to_V(c)
            (-29, -712/9, 19712/45, 0, -14/9, 364/45, 0, -4/9, 119/45)
            sage: from_V(to_V(c))
            7*a^2 - 10*a - 29
            sage: from_V(3*to_V(b))
            3*b
        """
        if base is not None and base is not QQ:
            raise ValueError("Absolute vector space base must be QQ")
        return self.free_module(QQ, *args, **kwds)

    def vector_space(self, *args, **kwds):
        r"""
        For a relative number field, ``vector_space()`` is
        deliberately not implemented, so that a user cannot confuse
        :meth:`~relative_vector_space` with :meth:`~absolute_vector_space`.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.vector_space()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field L you must use either L.relative_vector_space() or L.absolute_vector_space() as appropriate

        """
        raise NotImplementedError("For a relative number field L you must use either L.relative_vector_space() or L.absolute_vector_space() as appropriate")

    def absolute_base_field(self):
        r"""
        Return the base field of this relative extension, but viewed
        as an absolute field over `\QQ`.

        EXAMPLES::

            sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 3, x^3 + 2])
            sage: K
            Number Field in a with defining polynomial x^2 + 2 over its base field
            sage: K.base_field()
            Number Field in b with defining polynomial x^3 + 3 over its base field
            sage: K.absolute_base_field()[0]
            Number Field in a0 with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1
            sage: K.base_field().absolute_field('z')
            Number Field in z with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1
        """
        return self.__absolute_base_field

    @cached_method
    def _pari_rnfeq(self):
        """
        Return PARI data attached to this relative number field.

        OUTPUT:

        A 5-element PARI vector containing an absolute polynomial and
        further data needed for ``eltabstorel`` and ``eltreltoabs``,
        obtained from PARI's ``nf_rnfeq`` function.  This means that
        we do not have to initialize a full PARI ``rnf`` structure.

        TESTS::

            sage: K.<a> = NumberField(x^2 + 2)
            sage: x = polygen(K)
            sage: L.<b> = K.extension(x^5 + 2*a)
            sage: L._pari_rnfeq()
            [x^10 + 8, -1/2*x^5, 0, y^2 + 2, x^5 + 2*y]
            sage: x = polygen(ZZ)
            sage: NumberField(x^10 + 8, 'a').is_isomorphic(L)
            True

        Initialization is lazy enough to allow arithmetic in massive fields::

            sage: K.<a> = NumberField(x^10 + 2000*x + 100001)
            sage: x = polygen(K)
            sage: L.<b> = K.extension(x^10 + 2*a)
            sage: L._pari_rnfeq()
            [x^100 - 1024000*x^10 + 102401024, -1/2*x^10, 0, y^10 + 2000*y + 100001, x^10 + 2*y]
            sage: a + b
            b + a
            sage: b^100
            -2048000*a - 102401024
            sage: (-2*a)^10
            -2048000*a - 102401024
        """
        f = self.pari_absolute_base_polynomial()
        g = self.pari_relative_polynomial()
        return f._nf_rnfeq(g)

    @cached_method
    def _pari_nfzk(self):
        """
        Return PARI data needed for constructing relative number field
        elements from elements of the base field.

        TESTS::

            sage: K.<a> = NumberField(x^2 - 2)
            sage: L.<b> = K.extension(x^2 - 3)
            sage: L._pari_nfzk()
            [2, -x^3 + 9*x]
        """
        return self._pari_base_nf()._nf_nfzk(self._pari_rnfeq())

    @cached_method
    def _pari_relative_structure(self):
        r"""
        Return data relating the Sage and PARI relative polynomials.

        OUTPUT:

        Let `L` be this relative number field, let `K` be its base
        field, and let `f` be the defining polynomial of `L` over `K`.
        This method returns a triple ``(g, alpha, beta)``, where

        - ``g`` is the defining relative polynomial of the PARI
          ``rnf`` structure (see :meth:`pari_rnf`);

        - ``alpha`` is the image of `x \bmod f` under some isomorphism
          `\phi\colon K[x]/(f) \to K[x]/(g)`;

        - ``beta`` is the image of `x \bmod g` under the inverse
          isomorphism `\phi^{-1}\colon K[x]/(g) \to K[x]/(f)`.

        EXAMPLES::

        If the defining polynomials are monic and integral, the result
        satisfies ``g = f`` and ``alpha = beta = x``::

            sage: R.<x> = QQ[]
            sage: K.<a> = NumberField(x^2 + 1)
            sage: L.<b> = K.extension(x^2 - a)
            sage: L._pari_relative_structure()
            (Mod(1, y^2 + 1)*x^2 + Mod(-y, y^2 + 1),
             Mod(x, Mod(1, y^2 + 1)*x^2 + Mod(-y, y^2 + 1)),
             Mod(x, Mod(1, y^2 + 1)*x^2 + Mod(-y, y^2 + 1)))

        An example where the base field is defined by a monic integral
        polynomial, but the extension is not::

            sage: K.<a> = NumberField(x^2 + 1)
            sage: L.<b> = K.extension(x^2 - 1/2)
            sage: L._pari_relative_structure()
            (x^2 + Mod(-y, y^2 + 1),
             Mod(Mod(1/2*y - 1/2, y^2 + 1)*x, x^2 + Mod(-y, y^2 + 1)),
             Mod(Mod(-y - 1, y^2 + 1)*x, Mod(1, y^2 + 1)*x^2 + Mod(-1/2, y^2 + 1)))

        An example where both fields are defined by non-integral or
        non-monic polynomials::

            sage: K.<a> = NumberField(2*x^2 + 1)
            sage: L.<b> = K.extension(x^2 - 1/3)
            sage: L._pari_relative_structure()
            (x^2 + Mod(y, y^2 + 2)*x + 1,
             Mod(Mod(-1/3*y, y^2 + 2)*x + Mod(1/3, y^2 + 2), x^2 + Mod(y, y^2 + 2)*x + 1),
             Mod(Mod(3/2*y, y^2 + 2)*x + Mod(-1/2*y, y^2 + 2), Mod(1, y^2 + 2)*x^2 + Mod(-1/3, y^2 + 2)))

        Note that in the last example, the *absolute* defining
        polynomials is the same for Sage and PARI, even though this is
        not the case for the base field::

            sage: K._pari_absolute_structure()
            (y^2 + 2, Mod(1/2*y, y^2 + 2), Mod(2*y, y^2 + 1/2))
            sage: L._pari_absolute_structure()
            (y^4 + 4*y^2 + 1, Mod(y, y^4 + 4*y^2 + 1), Mod(y, y^4 + 4*y^2 + 1))
        """
        f = self.relative_polynomial()._pari_with_name('x')
        if f.pollead() == f.content().lift().content().denominator() == 1:
            g = f
            alpha = beta = f.variable().Mod(f)
        elif f.poldegree() == 1:
            # PARI's rnfpolredbest() does not always return a
            # polynomial with integral coefficients in this case.
            from sage.libs.pari.all import pari
            g = f.variable()
            alpha = -f[0]/f[1]
            beta = pari(0).Mod(f)
        else:
            g, alpha = self._pari_base_nf().rnfpolredbest(f, flag=1)
            beta = alpha.modreverse()
        return g, alpha, beta

    @cached_method
    def _gen_relative(self):
        r"""
        Return root of defining polynomial, which is a generator of
        the relative number field over the base.

        EXAMPLES::

            sage: k.<a> = NumberField(x^2+1); k
            Number Field in a with defining polynomial x^2 + 1
            sage: y = polygen(k)
            sage: m.<b> = k.extension(y^2+3); m
            Number Field in b with defining polynomial x^2 + 3 over its base field
            sage: c = m.gen(); c # indirect doctest
            b
            sage: c^2 + 3
            0

        An example where the defining polynomials are not monic or
        integral::

            sage: K.<a> = NumberField(3*x^2 + 1)
            sage: L.<b> = K.extension(x^2 - 1/2)
            sage: L._gen_relative()
            b
        """
        alpha = self._pari_relative_structure()[1].liftpol()
        return self._element_constructor_(self._pari_rnfeq()._eltreltoabs(alpha), check=False)

    @cached_method
    def pari_rnf(self):
        r"""
        Return the PARI relative number field object associated
        to this relative extension.

        EXAMPLES::

            sage: k.<a> = NumberField([x^4 + 3, x^2 + 2])
            sage: k.pari_rnf()
            [x^4 + 3, [364, -10*x^7 - 87*x^5 - 370*x^3 - 41*x], [108, 3], ...]
        """
        return self._pari_base_nf().rnfinit(self.pari_relative_polynomial())

    def pari_absolute_base_polynomial(self):
        r"""
        Return the PARI polynomial defining the absolute base field, in ``y``.

        EXAMPLES::

            sage: x = polygen(ZZ)
            sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 3]); K
            Number Field in a with defining polynomial x^2 + 2 over its base field
            sage: K.pari_absolute_base_polynomial()
            y^2 + 3
            sage: type(K.pari_absolute_base_polynomial())
            <class 'cypari2.gen.Gen'>
            sage: z = ZZ['z'].0
            sage: K.<a, b, c> = NumberField([z^2 + 2, z^2 + 3, z^2 + 5]); K
            Number Field in a with defining polynomial z^2 + 2 over its base field
            sage: K.pari_absolute_base_polynomial()
            y^4 + 16*y^2 + 4
            sage: K.base_field()
            Number Field in b with defining polynomial z^2 + 3 over its base field
            sage: len(QQ['y'](K.pari_absolute_base_polynomial()).roots(K.base_field()))
            4
            sage: type(K.pari_absolute_base_polynomial())
            <class 'cypari2.gen.Gen'>
        """
        abs_base, from_abs_base, to_abs_base = self.absolute_base_field()
        return abs_base.pari_polynomial('y')

    def pari_relative_polynomial(self):
        r"""
        Return the PARI relative polynomial associated to this number
        field.

        This is always a polynomial in x and y, suitable for PARI's
        rnfinit function.  Notice that if this is a relative extension
        of a relative extension, the base field is the absolute base
        field.

        EXAMPLES::

            sage: k.<i> = NumberField(x^2 + 1)
            sage: m.<z> = k.extension(k['w']([i,0,1]))
            sage: m
            Number Field in z with defining polynomial w^2 + i over its base field
            sage: m.pari_relative_polynomial()
            Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1)

            sage: l.<t> = m.extension(m['t'].0^2 + z)
            sage: l.pari_relative_polynomial()
            Mod(1, y^4 + 1)*x^2 + Mod(y, y^4 + 1)
        """
        return self._pari_relative_structure()[0]

    def number_of_roots_of_unity(self):
        """
        Return number of roots of unity in this relative field.

        EXAMPLES::

            sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] )
            sage: K.number_of_roots_of_unity()
            24
        """
        return self.absolute_field('a').number_of_roots_of_unity()

    def roots_of_unity(self):
        """
        Return all the roots of unity in this relative field, primitive or not.

        EXAMPLES::

            sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] )
            sage: rts = K.roots_of_unity()
            sage: len(rts)
            24
            sage: all(u in rts for u in [b*a, -b^2*a - b^2, b^3, -a, b*a + b])
            True
        """
        abs = self.absolute_field('a')
        from_abs, _ = abs.structure()
        return [from_abs(x) for x in abs.roots_of_unity()]

    def absolute_generator(self):
        r"""
        Return the chosen generator over `\QQ` for this relative number field.

        EXAMPLES::

            sage: y = polygen(QQ,'y')
            sage: k.<a> = NumberField([y^2 + 2, y^4 + 3])
            sage: g = k.absolute_generator(); g
            a0 - a1
            sage: g.minpoly()
            x^2 + 2*a1*x + a1^2 + 2
            sage: g.absolute_minpoly()
            x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
        """
        try:
            return self.__abs_gen
        except AttributeError:
            self.__abs_gen = self._element_class(self, QQ['x'].gen())
            return self.__abs_gen

    def absolute_field(self, names):
        """
        Return ``self`` as an absolute number field.

        INPUT:

        - ``names`` -- string; name of generator of the absolute field

        OUTPUT:

        An absolute number field `K` that is isomorphic to this field.

        Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where
        ``from_K`` is an isomorphism from `K` to ``self`` and ``to_K``
        is an isomorphism from ``self`` to `K`.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: L.<xyz> = K.absolute_field(); L
            Number Field in xyz with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
            sage: L.<c> = K.absolute_field(); L
            Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49

            sage: from_L, to_L = L.structure()
            sage: from_L
            Isomorphism map:
              From: Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
              To:   Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: from_L(c)
            a - b
            sage: to_L
            Isomorphism map:
              From: Number Field in a with defining polynomial x^4 + 3 over its base field
              To:   Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
            sage: to_L(a)
            -5/182*c^7 - 87/364*c^5 - 185/182*c^3 + 323/364*c
            sage: to_L(b)
            -5/182*c^7 - 87/364*c^5 - 185/182*c^3 - 41/364*c
            sage: to_L(a)^4
            -3
            sage: to_L(b)^2
            -2
        """
        return NumberField(self.absolute_polynomial(), names, structure=structure.AbsoluteFromRelative(self))

    def absolute_polynomial_ntl(self):
        """
        Return defining polynomial of this number field
        as a pair, an ntl polynomial and a denominator.

        This is used mainly to implement some internal arithmetic.

        EXAMPLES::

            sage: NumberField(x^2 + (2/3)*x - 9/17,'a').absolute_polynomial_ntl()
            ([-27 34 51], 51)
        """
        try:
            return (self.__abs_polynomial_ntl, self.__abs_denominator_ntl)
        except AttributeError:
            self.__abs_denominator_ntl = ntl.ZZ()
            den = self.absolute_polynomial().denominator()
            self.__abs_denominator_ntl.set_from_sage_int(ZZ(den))
            self.__abs_polynomial_ntl = ntl.ZZX((self.absolute_polynomial()*den).list())
        return (self.__abs_polynomial_ntl, self.__abs_denominator_ntl)

    @cached_method
    def absolute_polynomial(self):
        r"""
        Return the polynomial over `\QQ` that defines this field as an
        extension of the rational numbers.

        .. NOTE::

            The absolute polynomial of a relative number field is
            chosen to be equal to the defining polynomial of the
            underlying PARI absolute number field (it cannot be
            specified by the user).  In particular, it is always a
            monic polynomial with integral coefficients.  On the other
            hand, the defining polynomial of an absolute number field
            and the relative polynomial of a relative number field are
            in general different from their PARI counterparts.

        EXAMPLES::

            sage: k.<a, b> = NumberField([x^2 + 1, x^3 + x + 1]); k
            Number Field in a with defining polynomial x^2 + 1 over its base field
            sage: k.absolute_polynomial()
            x^6 + 5*x^4 - 2*x^3 + 4*x^2 + 4*x + 1

        An example comparing the various defining polynomials to their
        PARI counterparts::

            sage: k.<a, c> = NumberField([x^2 + 1/3, x^2 + 1/4])
            sage: k.absolute_polynomial()
            x^4 - x^2 + 1
            sage: k.pari_polynomial()
            x^4 - x^2 + 1
            sage: k.base_field().absolute_polynomial()
            x^2 + 1/4
            sage: k.pari_absolute_base_polynomial()
            y^2 + 1
            sage: k.relative_polynomial()
            x^2 + 1/3
            sage: k.pari_relative_polynomial()
            x^2 + Mod(y, y^2 + 1)*x - 1
        """
        return QQ['x'](self._pari_rnfeq()[0])

    def relative_polynomial(self):
        """
        Return the defining polynomial of this relative number field over its base field.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.relative_polynomial()
            x^2 + x + 1

        Use absolute polynomial for a polynomial that defines the absolute
        extension.::

            sage: K.absolute_polynomial()
            x^6 + 3*x^5 + 8*x^4 + 9*x^3 + 7*x^2 + 6*x + 3
        """
        return self.__relative_polynomial

    def defining_polynomial(self):
        """
        Return the defining polynomial of this relative number field.

        This is exactly the same as ``relative_polynomial()``.

        EXAMPLES::

            sage: C.<z> = CyclotomicField(5)
            sage: PC.<X> = C[]
            sage: K.<a> = C.extension(X^2 + X + z); K
            Number Field in a with defining polynomial X^2 + X + z over its base field
            sage: K.defining_polynomial()
            X^2 + X + z
        """
        return self.relative_polynomial()

    def polynomial(self):
        """
        For a relative number field, ``polynomial()`` is deliberately
        not implemented.  Either :meth:`~relative_polynomial` or
        :meth:`~absolute_polynomial` must be used.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.polynomial()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field L you must use either L.relative_polynomial() or L.absolute_polynomial() as appropriate
        """
        raise NotImplementedError("For a relative number field L you must use either L.relative_polynomial() or L.absolute_polynomial() as appropriate")

    def base_field(self):
        """
        Return the base field of this relative number field.

        EXAMPLES::

            sage: k.<a> = NumberField([x^3 + x + 1])
            sage: R.<z> = k[]
            sage: L.<b> = NumberField(z^3 + a)
            sage: L.base_field()
            Number Field in a with defining polynomial x^3 + x + 1
            sage: L.base_field() is k
            True

        This is very useful because the print representation of
        a relative field doesn't describe the base field.::

            sage: L
            Number Field in b with defining polynomial z^3 + a over its base field
        """
        return self.__base_field

    def base_ring(self):
        """
        This is exactly the same as base_field.

        EXAMPLES::

            sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1])
            sage: k.base_ring()
            Number Field in a1 with defining polynomial x^3 + x + 1
            sage: k.base_field()
            Number Field in a1 with defining polynomial x^3 + x + 1
        """
        return self.base_field()

    def embeddings(self, K):
        r"""
        Compute all field embeddings of the relative number field self
        into the field `K` (which need not even be a number field,
        e.g., it could be the complex numbers). This will return an
        identical result when given `K` as input again.

        If possible, the most natural embedding of self into `K`
        is put first in the list.

        INPUT:

        - ``K`` -- a field

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^3 - 2, x^2+1])
            sage: f = K.embeddings(ComplexField(58)); f
            [
            Relative number field morphism:
              From: Number Field in a with defining polynomial x^3 - 2 over its base field
              To:   Complex Field with 58 bits of precision
              Defn: a |--> -0.62996052494743676 - 1.0911236359717214*I
                    b |--> -1.9428902930940239e-16 + 1.0000000000000000*I,
            ...
            Relative number field morphism:
              From: Number Field in a with defining polynomial x^3 - 2 over its base field
              To:   Complex Field with 58 bits of precision
              Defn: a |--> 1.2599210498948731
                    b |--> -0.99999999999999999*I
            ]
            sage: f[0](a)^3
            2.0000000000000002 - 8.6389229103644993e-16*I
            sage: f[0](b)^2
            -1.0000000000000001 - 3.8857805861880480e-16*I
            sage: f[0](a+b)
            -0.62996052494743693 - 0.091123635971721295*I
        """
        try:
            # this should be concordant with automorphisms
            return self.__embeddings[K]
        except AttributeError:
            self.__embeddings = {}
        except KeyError:
            pass
        L = self.absolute_field('a')
        E = L.embeddings(K)
        v = [self.hom(f, K) for f in E]

        # If there is an embedding that preserves variable names
        # then it is most natural, so we put it first.
        put_natural_embedding_first(v)

        self.__embeddings[K] = Sequence(v, cr=v!=[], immutable=True, check=False, universe=self.Hom(K))
        return self.__embeddings[K]

    def automorphisms(self):
        r"""
        Compute all Galois automorphisms of self over the base field.  This is
        different than computing the embeddings of self into self; there,
        automorphisms that do not fix the base field are considered.

        EXAMPLES::

            sage: K.<a, b> = NumberField([x^2 + 10000, x^2 + x + 50]); K
            Number Field in a with defining polynomial x^2 + 10000 over its base field
            sage: K.automorphisms()
            [
            Relative number field endomorphism of Number Field in a with defining polynomial x^2 + 10000 over its base field
              Defn: a |--> a
                    b |--> b,
            Relative number field endomorphism of Number Field in a with defining polynomial x^2 + 10000 over its base field
              Defn: a |--> -a
                    b |--> b
            ]
            sage: rho, tau = K.automorphisms()
            sage: tau(a)
            -a
            sage: tau(b) == b
            True

            sage: L.<b, a> = NumberField([x^2 + x + 50, x^2 + 10000, ]); L
            Number Field in b with defining polynomial x^2 + x + 50 over its base field
            sage: L.automorphisms()
            [
            Relative number field endomorphism of Number Field in b with defining polynomial x^2 + x + 50 over its base field
              Defn: b |--> b
                    a |--> a,
            Relative number field endomorphism of Number Field in b with defining polynomial x^2 + x + 50 over its base field
              Defn: b |--> -b - 1
                    a |--> a
            ]
            sage: rho, tau = L.automorphisms()
            sage: tau(a) == a
            True
            sage: tau(b)
            -b - 1

            sage: PQ.<X> = QQ[]
            sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
            sage: PF.<Y> = F[]
            sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
            sage: K.automorphisms()
            [
            Relative number field endomorphism of Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c |--> c
                    a |--> a
                    b |--> b,
            Relative number field endomorphism of Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c |--> -c
                    a |--> a
                    b |--> b
            ]
        """
        try:
            return self.__automorphisms
        except AttributeError:
            pass

        L = self.absolute_field('a')
        L_into_self, self_into_L = L.structure()
        aas = L.automorphisms() # absolute automorphisms

        a = self_into_L(self.gen())
        abs_base_gens = [self_into_L(_) for _ in self.base_field().gens()]
        v = sorted([ self.hom([ L_into_self(aa(a)) ]) for aa in aas if all(aa(g) == g for g in abs_base_gens) ])
        put_natural_embedding_first(v)
        self.__automorphisms = Sequence(v, cr = (v != []), immutable=True,
                                        check=False, universe=self.Hom(self))
        return self.__automorphisms

    def places(self, all_complex=False, prec=None):
        """
        Return the collection of all infinite places of self.

        By default, this returns the set of real places as
        homomorphisms into RIF first, followed by a choice of one of
        each pair of complex conjugate homomorphisms into CIF.

        On the other hand, if prec is not None, we simply return places
        into RealField(prec) and ComplexField(prec) (or RDF, CDF if
        prec=53).

        There is an optional flag all_complex, which defaults to False. If
        all_complex is True, then the real embeddings are returned as
        embeddings into CIF instead of RIF.

        EXAMPLES::

            sage: L.<b, c> = NumberFieldTower([x^2 - 5, x^3 + x + 3])
            sage: L.places()
            [Relative number field morphism:
            From: Number Field in b with defining polynomial x^2 - 5 over its base field
            To:   Real Field with 106 bits of precision
            Defn: b |--> -2.236067977499789696409173668937
            c |--> -1.213411662762229634132131377426,
            Relative number field morphism:
            From: Number Field in b with defining polynomial x^2 - 5 over its base field
            To:   Real Field with 106 bits of precision
            Defn: b |--> 2.236067977499789696411548005367
            c |--> -1.213411662762229634130492421800,
            Relative number field morphism:
            From: Number Field in b with defining polynomial x^2 - 5 over its base field
            To:   Complex Field with 53 bits of precision
            Defn: b |--> -2.23606797749979 ...e-1...*I
            c |--> 0.606705831381... - 1.45061224918844*I,
            Relative number field morphism:
            From: Number Field in b with defining polynomial x^2 - 5 over its base field
            To:   Complex Field with 53 bits of precision
            Defn: b |--> 2.23606797749979 - 4.44089209850063e-16*I
            c |--> 0.606705831381115 - 1.45061224918844*I]
        """
        L = self.absolute_field('a')
        pl = L.places(all_complex, prec)
        return [self.hom(p, p.codomain()) for p in pl]

    def absolute_different(self):
        r"""
        Return the absolute different of this relative number field `L`, as an
        ideal of `L`. To get the relative different of `L/K`, use
        ``L.relative_different()``.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2 + 1)
            sage: t = K['t'].gen()
            sage: L.<b> = K.extension(t^4 - i)
            sage: L.absolute_different()
            Fractional ideal (8)
        """
        abs = self.absolute_field('a')
        from_abs = abs.structure()[0]
        return self.ideal([from_abs(g) for g in abs.different().gens()])

    def relative_different(self):
        r"""
        Return the relative different of this extension `L/K` as
        an ideal of `L`.  If you want the absolute different of
        `L/\QQ`, use ``L.absolute_different()``.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2 + 1)
            sage: PK.<t> = K[]
            sage: L.<a> = K.extension(t^4  - i)
            sage: L.relative_different()
            Fractional ideal (4)
        """
        I = self.absolute_different()
        J = self.ideal(self.base_field().absolute_different().gens())
        return  I/J

    def different(self):
        """
        The different, unqualified, of a relative number field is deliberately
        not implemented, so that a user cannot mistake the absolute different
        for the relative different, or vice versa.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.different()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field you must use relative_different or absolute_different as appropriate
        """
        raise NotImplementedError("For a relative number field you must use relative_different or absolute_different as appropriate")

    def absolute_discriminant(self, v=None):
        r"""
        Return the absolute discriminant of this relative number field
        or if ``v`` is specified, the determinant of the trace pairing
        on the elements of the list ``v``.

        INPUT:

        - ``v`` (optional) -- list of element of this relative number field.

        OUTPUT: Integer if ``v`` is omitted, and Rational otherwise.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2 + 1)
            sage: t = K['t'].gen()
            sage: L.<b> = K.extension(t^4 - i)
            sage: L.absolute_discriminant()
            16777216
            sage: L.absolute_discriminant([(b + i)^j for j in range(8)])
            61911970349056
        """
        abs = self.absolute_field('a')
        if v is not None:
            to_abs = abs.structure()[1]
            v = [to_abs(x) for x in v]
        return abs.discriminant(v=v)

    def relative_discriminant(self):
        r"""
        Return the relative discriminant of this extension `L/K` as an ideal of
        `K`. If you want the (rational) discriminant of `L/\QQ`, use e.g.
        ``L.absolute_discriminant()``.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2 + 1)
            sage: t = K['t'].gen()
            sage: L.<b> = K.extension(t^4 - i)
            sage: L.relative_discriminant()
            Fractional ideal (256)
            sage: PQ.<X> = QQ[]
            sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
            sage: PF.<Y> = F[]
            sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
            sage: K.relative_discriminant() == F.ideal(4*b)
            True

        TESTS:

        Number fields defined by non-monic and non-integral
        polynomials are supported (:trac:`252`)::

            sage: K.<a> = NumberField(x^2 + 1/2)
            sage: L.<b> = K.extension(x^2 - 1/2)
            sage: L.relative_discriminant()
            Fractional ideal (2)
        """
        base = self.base_field()
        nf = base.pari_nf()
        D, d = nf.rnfdisc(self.pari_relative_polynomial())
        return base.ideal(nf.idealhnf(D))

    def discriminant(self):
        """
        The discriminant, unqualified, of a relative number field is deliberately
        not implemented, so that a user cannot mistake the absolute discriminant
        for the relative discriminant, or vice versa.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.discriminant()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate
        """
        raise NotImplementedError("For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate")

    def disc(self):
        """
        The discriminant, unqualified, of a relative number field is deliberately
        not implemented, so that a user cannot mistake the absolute discriminant
        for the relative discriminant, or vice versa.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.disc()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate
        """
        raise NotImplementedError("For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate")

    def order(self, *gens, **kwds):
        r"""
        Return the order with given ring generators in the maximal
        order of this number field.

        INPUT:

        - ``gens`` -- list of elements of self; if no generators are given, just
          returns the cardinality of this number field (oo) for consistency.
        - ``check_is_integral`` -- bool (default: True), whether to check that each
          generator is integral.
        - ``check_rank`` -- bool (default: True), whether to check that the ring
          generated by gens is of full rank.
        - ``allow_subfield`` -- bool (default: False), if True and the generators
          do not generate an order, i.e., they generate a subring of smaller
          rank, instead of raising an error, return an order in a smaller
          number field.

        The check_is_integral and check_rank inputs must be given as
        explicit keyword arguments.

        EXAMPLES::

            sage: P.<a,b,c> = QQ[2^(1/2), 2^(1/3), 3^(1/2)]
            sage: R = P.order([a,b,c]); R
            Relative Order in Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field

        The base ring of an order in a relative extension is still `\ZZ`.::

            sage: R.base_ring()
            Integer Ring

        One must give enough generators to generate a ring of finite index
        in the maximal order::

            sage: P.order([a,b])
            Traceback (most recent call last):
            ...
            ValueError: the rank of the span of gens is wrong
        """
        if len(gens) == 0:
            return NumberField_generic.order(self)
        if len(gens) == 1 and isinstance(gens[0], (list, tuple)):
            gens = gens[0]
        gens = [self(x) for x in gens]
        return relative_order_from_ring_generators(gens, **kwds)

    def is_free(self, proof=None):
        r"""
        Determine whether or not `L/K` is free (i.e. if `\mathcal{O}_L` is
        a free `\mathcal{O}_K`-module).

        INPUT:

        - ``proof`` -- default: True

        EXAMPLES::

            sage: x = polygen(QQ)
            sage: K.<a> = NumberField(x^2+6)
            sage: x = polygen(K)
            sage: L.<b> = K.extension(x^2 + 3)    ## extend by x^2+3
            sage: L.is_free()
            False
        """
        proof = proof_flag(proof)
        base_bnf = self._pari_base_bnf(proof)
        if base_bnf.rnfisfree(self.pari_relative_polynomial()) == 1:
            return True
        return False

    def _factor_univariate_polynomial(self, poly, **kwargs):
        """
        Factorisation of univariate polynomials over relative number fields.

        This is called by the ``factor`` method of univariate polynomials.

        EXAMPLES::

            sage: K.<i> = NumberField(x**2+1)
            sage: L.<sqrt2> = K.extension(x*x-2)
            sage: x = polygen(L,'x')
            sage: factor(x**2+8)  # indirect doctest
            (x + 2*i*sqrt2) * (x - 2*i*sqrt2)
        """
        M = self.absolute_field('a')
        from_M, to_M = M.structure()
        g = M['x']([to_M(x) for x in poly.list()])
        F = g.factor()
        S = poly.parent()
        v = [(S([from_M(x) for x in f.list()]), e) for f, e in F]
        return Factorization(v, from_M(F.unit()))

    def lift_to_base(self, element):
        """
        Lift an element of this extension into the base field if possible,
        or raise a ValueError if it is not possible.

        EXAMPLES::

            sage: x = polygen(ZZ)
            sage: K.<a> = NumberField(x^3 - 2)
            sage: R.<y> = K[]
            sage: L.<b> = K.extension(y^2 - a)
            sage: L.lift_to_base(b^4)
            a^2
            sage: L.lift_to_base(b^6)
            2
            sage: L.lift_to_base(355/113)
            355/113
            sage: L.lift_to_base(b)
            Traceback (most recent call last):
            ...
            ValueError: The element b is not in the base field

        TESTS:

        Number fields defined by non-monic and non-integral
        polynomials are supported (:trac:`252`)::

            sage: R.<x> = QQ[]
            sage: K.<a> = NumberField(x^2 + 1/2)
            sage: L.<b> = K.extension(x^2 - a/2)
            sage: L.lift_to_base(b^2)
            1/2*a
        """
        # Convert the element to a PARI polynomial with t_POLMOD
        # coefficients representing elements of the base field.
        r = self._pari_rnfeq()._eltabstorel_lift(self(element)._pari_polynomial('x'))
        # Lift the coefficients and call simplify() to make PARI check
        # which variables really appear in the resulting polynomial
        # (otherwise we always have a polynomial in two variables even
        # though not all variables actually occur).
        r = r.lift().simplify()

        # Special case: check whether the result is simply an integer or rational
        if r.type() in ["t_INT", "t_FRAC"]:
            return self.base_field()(r)
        # Now we should have a polynomial in the variable y.
        # Otherwise we're not in the base field.
        if r.type() != "t_POL" or str(r.variable()) != 'y':
            raise ValueError("The element %s is not in the base field"%element)
        return self.base_field()(r, check=False)

    def relativize(self, alpha, names):
        r"""
        Given an element in self or an embedding of a subfield into self,
        return a relative number field `K` isomorphic to self that is relative
        over the absolute field `\QQ(\alpha)` or the domain of `\alpha`, along
        with isomorphisms from `K` to self and from self to `K`.

        INPUT:

        - ``alpha`` -- an element of self, or an embedding of a subfield into self
        - ``names`` -- name of generator for output field `K`.

        OUTPUT: `K` -- a relative number field

        Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where
        ``from_K`` is an isomorphism from `K` to self and ``to_K`` is
        an isomorphism from self to `K`.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: L.<z,w> = K.relativize(a^2)
            sage: z^2
            z^2
            sage: w^2
            -3
            sage: L
            Number Field in z with defining polynomial x^4 + (-2*w + 4)*x^2 + 4*w + 1 over its base field
            sage: L.base_field()
            Number Field in w with defining polynomial x^2 + 3

        Now suppose we have `K` below `L` below `M`::

            sage: M = NumberField(x^8 + 2, 'a'); M
            Number Field in a with defining polynomial x^8 + 2
            sage: L, L_into_M, _ = M.subfields(4)[0]; L
            Number Field in a0 with defining polynomial x^4 + 2
            sage: K, K_into_L, _ = L.subfields(2)[0]; K
            Number Field in a0_0 with defining polynomial x^2 + 2
            sage: K_into_M = L_into_M * K_into_L

            sage: L_over_K = L.relativize(K_into_L, 'c'); L_over_K
            Number Field in c with defining polynomial x^2 + a0_0 over its base field
            sage: L_over_K_to_L, L_to_L_over_K = L_over_K.structure()
            sage: M_over_L_over_K = M.relativize(L_into_M * L_over_K_to_L, 'd'); M_over_L_over_K
            Number Field in d with defining polynomial x^2 + c over its base field
            sage: M_over_L_over_K.base_field() is L_over_K
            True

        Test relativizing a degree 6 field over its degree 2 and degree 3
        subfields, using both an explicit element::

            sage: K.<a> = NumberField(x^6 + 2); K
            Number Field in a with defining polynomial x^6 + 2
            sage: K2, K2_into_K, _ = K.subfields(2)[0]; K2
            Number Field in a0 with defining polynomial x^2 + 2
            sage: K3, K3_into_K, _ = K.subfields(3)[0]; K3
            Number Field in a0 with defining polynomial x^3 - 2

        Here we explicitly relativize over an element of K2 (not the
        generator)::

            sage: L = K.relativize(K3_into_K, 'b'); L
            Number Field in b with defining polynomial x^2 + a0 over its base field
            sage: L_to_K, K_to_L = L.structure()
            sage: L_over_K2 = L.relativize(K_to_L(K2_into_K(K2.gen() + 1)), 'c'); L_over_K2
            Number Field in c0 with defining polynomial x^3 - c1 + 1 over its base field
            sage: L_over_K2.base_field()
            Number Field in c1 with defining polynomial x^2 - 2*x + 3

        Here we use a morphism to preserve the base field information::

            sage: K2_into_L = K_to_L * K2_into_K
            sage: L_over_K2 = L.relativize(K2_into_L, 'c'); L_over_K2
            Number Field in c with defining polynomial x^3 - a0 over its base field
            sage: L_over_K2.base_field() is K2
            True
        """
        K = self.absolute_field('a')
        from_K, to_K = K.structure()

        if is_Map(alpha):
            # alpha is an embedding of a subfield into self; compose to get an
            # embedding of a subfield into the absolute field
            beta = to_K * alpha
        else:
            # alpha is an element coercible into self
            beta = to_K(alpha)

        L = K.relativize(beta, names)
        return K.relativize(beta, names, structure=structure.RelativeFromRelative(L))

    def uniformizer(self, P, others = "positive"):
        """
        Returns an element of self with valuation 1 at the prime ideal P.

        INPUT:


        -  ``self`` - a number field

        -  ``P`` - a prime ideal of self

        -  ``others`` - either "positive" (default), in which
           case the element will have non-negative valuation at all other
           primes of self, or "negative", in which case the element will have
           non-positive valuation at all other primes of self.


        .. note::

           When P is principal (e.g. always when self has class number
           one) the result may or may not be a generator of P!

        EXAMPLES::

            sage: K.<a, b> = NumberField([x^2 + 23, x^2 - 3])
            sage: P = K.prime_factors(5)[0]; P
            Fractional ideal (5, 1/2*a + b - 5/2)
            sage: u = K.uniformizer(P)
            sage: u.valuation(P)
            1
            sage: (P, 1) in K.factor(u)
            True
        """
        if not is_NumberFieldIdeal(P):
            P = self.ideal(P)
        if not P.is_maximal():
            raise ValueError("P (=%s) must be a nonzero prime."%P)
        abs = self.absolute_field('a')
        from_abs = abs.structure()[0]
        return from_abs(abs.uniformizer(P.absolute_ideal(), others=others))


def NumberField_relative_v1(base_field, poly, name, latex_name, canonical_embedding=None):
    """
    Used for unpickling old pickles.

    EXAMPLES::

        sage: from sage.rings.number_field.number_field_rel import NumberField_relative_v1
        sage: R.<x> = CyclotomicField(3)[]
        sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a')
        Number Field in a with defining polynomial x^2 + 7 over its base field
    """
    return NumberField(poly.change_ring(base_field), name, check=False,
                       embedding=canonical_embedding, latex_name=latex_name)

NumberField_extension_v1 = NumberField_relative_v1  # historical reasons only

r"""
Relative Number Fields

AUTHORS:

- <NAME> (2004, 2005): initial version
- <NAME> (2006-05-12): added support for relative extensions
- <NAME> (2007-09-04): major rewrite and documentation
- <NAME> (2008-10): specified embeddings into ambient fields
- <NAME> (2009-01): modernize coercion implementation
- <NAME> (2012-08): added is_CM_extension
- <NAME> (2014-04): absolute number fields are unique parents

This example follows one in the Magma reference manual::

    sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000)
    sage: z = y^5/11; z
    420/11*y^3 - 40000/11*y
    sage: R.<y> = PolynomialRing(K)
    sage: f = y^2 + y + 1
    sage: L.<a> = K.extension(f); L
    Number Field in a with defining polynomial y^2 + y + 1 over its base field
    sage: KL.<b> = NumberField([x^4 - 420*x^2 + 40000, x^2 + x + 1]); KL
    Number Field in b0 with defining polynomial x^4 - 420*x^2 + 40000 over its base field

We do some arithmetic in a tower of relative number fields::

    sage: K.<cuberoot2> = NumberField(x^3 - 2)
    sage: L.<cuberoot3> = K.extension(x^3 - 3)
    sage: S.<sqrt2> = L.extension(x^2 - 2)
    sage: S
    Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
    sage: sqrt2 * cuberoot3
    cuberoot3*sqrt2
    sage: (sqrt2 + cuberoot3)^5
    (20*cuberoot3^2 + 15*cuberoot3 + 4)*sqrt2 + 3*cuberoot3^2 + 20*cuberoot3 + 60
    sage: cuberoot2 + cuberoot3
    cuberoot3 + cuberoot2
    sage: cuberoot2 + cuberoot3 + sqrt2
    sqrt2 + cuberoot3 + cuberoot2
    sage: (cuberoot2 + cuberoot3 + sqrt2)^2
    (2*cuberoot3 + 2*cuberoot2)*sqrt2 + cuberoot3^2 + 2*cuberoot2*cuberoot3 + cuberoot2^2 + 2
    sage: cuberoot2 + sqrt2
    sqrt2 + cuberoot2
    sage: a = S(cuberoot2); a
    cuberoot2
    sage: a.parent()
    Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field

WARNING: Doing arithmetic in towers of relative fields that depends on
canonical coercions is currently VERY SLOW.  It is much better to
explicitly coerce all elements into a common field, then do arithmetic
with them there (which is quite fast).

TESTS::

    sage: y = polygen(QQ,'y'); K.<beta> = NumberField([y^3 - 3, y^2 - 2])
    sage: K(y^10)
    27*beta0
    sage: beta^10
    27*beta0
"""
# ****************************************************************************
#       Copyright (C) 2004-2009 <NAME> <<EMAIL>>
#                     2014-2022 <NAME> <<EMAIL>>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  https://www.gnu.org/licenses/
# ****************************************************************************

import sage.libs.ntl.all as ntl

from sage.categories.map import is_Map
from sage.structure.sequence import Sequence

import sage.structure.parent_gens

from . import maps
from . import structure

from sage.misc.latex import latex
from sage.misc.cachefunc import cached_method
from sage.structure.factorization import Factorization
import sage.rings.polynomial.polynomial_element as polynomial_element
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

from . import number_field_element
import sage.rings.number_field.number_field_ideal_rel
from .number_field_ideal import is_NumberFieldIdeal
from .number_field import (NumberField, NumberField_generic,
    put_natural_embedding_first, proof_flag,
    is_NumberFieldHomsetCodomain)
from sage.rings.number_field.number_field_base import is_NumberField
from sage.rings.number_field.order import (RelativeOrder, is_NumberFieldOrder,
                                           relative_order_from_ring_generators)
from sage.rings.number_field.morphism import RelativeNumberFieldHomomorphism_from_abs
from sage.libs.pari.all import pari_gen

from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ


def is_RelativeNumberField(x):
    r"""
    Return ``True`` if `x` is a relative number field.

    EXAMPLES::

        sage: from sage.rings.number_field.number_field_rel import is_RelativeNumberField
        sage: is_RelativeNumberField(NumberField(x^2+1,'a'))
        False
        sage: k.<a> = NumberField(x^3 - 2)
        sage: l.<b> = k.extension(x^3 - 3); l
        Number Field in b with defining polynomial x^3 - 3 over its base field
        sage: is_RelativeNumberField(l)
        True
        sage: is_RelativeNumberField(QQ)
        False
    """
    return isinstance(x, NumberField_relative)


class NumberField_relative(NumberField_generic):
    """
    INPUT:

    - ``base`` -- the base field

    - ``polynomial`` -- a polynomial which must be defined in the ring `K[x]`,
      where `K` is the base field.

    - ``name`` -- a string, the variable name

    - ``latex_name`` -- a string or ``None`` (default: ``None``), variable name
      for latex printing

    - ``check`` -- a boolean (default: ``True``), whether to check
      irreducibility of ``polynomial``

    - ``embedding`` -- currently not supported, must be ``None``

    - ``structure`` -- an instance of :class:`structure.NumberFieldStructure`
      or ``None`` (default: ``None``), provides additional information about
      this number field, e.g., the absolute number field from which it was
      created

    EXAMPLES::

        sage: K.<a> = NumberField(x^3 - 2)
        sage: t = polygen(K)
        sage: L.<b> = K.extension(t^2+t+a); L
        Number Field in b with defining polynomial x^2 + x + a over its base field

    TESTS::

        sage: Z = var('Z')
        sage: K.<w> = NumberField(Z^3 + Z + 1)
        sage: L.<z> = K.extension(Z^3 + 2)
        sage: K = loads(dumps(L))
        sage: K
        Number Field in z with defining polynomial Z^3 + 2 over its base field
        sage: L is K
        True

        sage: M.<u,v> = L.change_names()
        sage: M.structure()
        (Isomorphism given by variable name change map:
          From: Number Field in u with defining polynomial x^3 + 2 over its base field
          To:   Number Field in z with defining polynomial Z^3 + 2 over its base field,
         Isomorphism given by variable name change map:
          From: Number Field in z with defining polynomial Z^3 + 2 over its base field
          To:   Number Field in u with defining polynomial x^3 + 2 over its base field)
        sage: loads(dumps(M)) is M
        True
    """
    def __init__(self, base, polynomial, name,
                 latex_name=None, names=None, check=True, embedding=None, structure=None):
        r"""
        Initialization.

        EXAMPLES::

            sage: K.<x> = CyclotomicField(5)[]
            sage: W.<a> = NumberField(x^2 + 1)
            sage: W
            Number Field in a with defining polynomial x^2 + 1 over its base field
            sage: type(W)
            <class 'sage.rings.number_field.number_field_rel.NumberField_relative_with_category'>

        Test that check=False really skips the test::

            sage: W.<a> = NumberField(K.cyclotomic_polynomial(5), check=False)
            sage: W
            Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1 over its base field

        A relative extension of a relative extension::

            sage: x = polygen(ZZ)
            sage: k.<a0,a1> = NumberField([x^2 + 2, x^2 + 1])
            sage: l.<b> = k.extension(x^2 + 3)
            sage: l
            Number Field in b with defining polynomial x^2 + 3 over its base field
            sage: l.base_field()
            Number Field in a0 with defining polynomial x^2 + 2 over its base field
            sage: l.base_field().base_field()
            Number Field in a1 with defining polynomial x^2 + 1

        Non-monic and non-integral polynomials are supported (:trac:`252`)::

            sage: l.<b> = k.extension(5*x^2 + 3); l
            Number Field in b with defining polynomial 5*x^2 + 3 over its base field
            sage: l.pari_rnf()
            [x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4), ..., y^4 + 6*y^2 + 1, x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4)], [0, 0]]
            sage: b
            b

            sage: l.<b> = k.extension(x^2 + 3/5); l
            Number Field in b with defining polynomial x^2 + 3/5 over its base field
            sage: l.pari_rnf()
            [x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4), ..., y^4 + 6*y^2 + 1, x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4)], [0, 0]]
            sage: b
            b

            sage: l.<b> = k.extension(x - 1/a0); l
            Number Field in b with defining polynomial x + 1/2*a0 over its base field
            sage: l.pari_rnf()
            [x, [4, -x^3 - x^2 - 7*x - 3, -x^3 + x^2 - 7*x + 3, 2*x^3 + 10*x], ..., [x^4 + 6*x^2 + 1, -x, -1, y^4 + 6*y^2 + 1, x], [0, 0]]
            sage: b
            -1/2*a0

        TESTS:

        Test that irreducibility testing is working::

            sage: x = polygen(ZZ)
            sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 3])
            sage: K.<a> = NumberField(x^2 + 2)
            sage: x = polygen(K)
            sage: L.<b> = K.extension(x^3 + 3*a)

            sage: (x^3 + 2*a).factor()
            (x - a) * (x^2 + a*x - 2)
            sage: L.<b> = K.extension(x^3 + 2*a)
            Traceback (most recent call last):
            ...
            ValueError: defining polynomial (x^3 + 2*a) must be irreducible
            sage: (x^2 + 2).factor()
            (x - a) * (x + a)
            sage: L.<b> = K.extension(x^2 + 2)
            Traceback (most recent call last):
            ...
            ValueError: defining polynomial (x^2 + 2) must be irreducible
            sage: L.<b> = K.extension(x^2 + 2)
            Traceback (most recent call last):
            ...
            ValueError: defining polynomial (x^2 + 2) must be irreducible

        Error checks::

            sage: x = polygen(ZZ)
            sage: K.<a> = NumberField(x^2 + 1)
            sage: K.extension(x^2 + 2, 'a')
            Traceback (most recent call last):
            ...
            ValueError: base field and extension cannot have the same name 'a'
        """
        if embedding is not None:
            raise NotImplementedError("Embeddings not implemented for relative number fields")
        if names is not None:
            name = names
        if not is_NumberField(base):
            raise TypeError("base (=%s) must be a number field"%base)
        if not isinstance(polynomial, polynomial_element.Polynomial):
            try:
                polynomial = polynomial.polynomial(base)
            except (AttributeError, TypeError):
                raise TypeError("polynomial (=%r) must be a polynomial" % polynomial)
        if name == base.variable_name():
            raise ValueError("base field and extension cannot have the same name %r" % name)
        if polynomial.parent().base_ring() != base:
            polynomial = polynomial.change_ring(base)
            #raise ValueError, "The polynomial must be defined over the base field"

        # Generate the nf and bnf corresponding to the base field
        # defined as polynomials in y, e.g. for rnfisfree

        # Convert the polynomial defining the base field into a
        # polynomial in y to satisfy PARI's ordering requirements.

        if base.is_relative():
            abs_base = base.absolute_field(name + '0')
            from_abs_base, to_abs_base = abs_base.structure()
        else:
            abs_base = base
            from_abs_base = maps.IdentityMap(base)
            to_abs_base = maps.IdentityMap(base)

        self.__absolute_base_field = abs_base, from_abs_base, to_abs_base
        self.__base_field = base
        self.__relative_polynomial = polynomial
        self._element_class = number_field_element.NumberFieldElement_relative

        if check and not self.pari_relative_polynomial().polisirreducible():
            raise ValueError("defining polynomial (%s) must be irreducible"%polynomial)

        names = (name,) + base.variable_names()
        self._assign_names(tuple(names), normalize=False)

        NumberField_generic.__init__(self, self.absolute_polynomial(), name=None,
                                     latex_name=latex_name, check=False,
                                     embedding=embedding, structure=structure)

        self._zero_element = self(0)
        self._one_element =  self(1)

    def change_names(self, names):
        r"""
        Return relative number field isomorphic to self but with the
        given generator names.

        INPUT:

        - ``names`` -- number of names should be at most the number of
          generators of self, i.e., the number of steps in the tower
          of relative fields.

        Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where
        from_K is an isomorphism from `K` to self and ``to_K`` is an
        isomorphism from self to `K`.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: L.<c,d> = K.change_names()
            sage: L
            Number Field in c with defining polynomial x^4 + 3 over its base field
            sage: L.base_field()
            Number Field in d with defining polynomial x^2 + 2

        An example with a 3-level tower::

            sage: K.<a,b,c> = NumberField([x^2 + 17, x^2 + x + 1, x^3 - 2]); K
            Number Field in a with defining polynomial x^2 + 17 over its base field
            sage: L.<m,n,r> = K.change_names()
            sage: L
            Number Field in m with defining polynomial x^2 + 17 over its base field
            sage: L.base_field()
            Number Field in n with defining polynomial x^2 + x + 1 over its base field
            sage: L.base_field().base_field()
            Number Field in r with defining polynomial x^3 - 2

        And a more complicated example::

            sage: PQ.<X> = QQ[]
            sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
            sage: PF.<Y> = F[]
            sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
            sage: L.<m, n, r> = K.change_names(); L
            Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field
            sage: L.structure()
            (Isomorphism given by variable name change map:
              From: Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field,
             Isomorphism given by variable name change map:
              From: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              To:   Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field)
        """
        if len(names) == 0:
            names = self.variable_names()
        elif isinstance(names, str):
            names = names.split(',')
        K = self.base_field().change_names(tuple(names[1:]))
        to_K = K.structure()[1]
        old_poly = self.relative_polynomial()
        new_poly = PolynomialRing(K, 'x')([to_K(c) for c in old_poly])
        return K.extension(new_poly, names=names[0], structure=structure.NameChange(self))

    def subfields(self, degree=0, name=None):
        """
        Return all subfields of this relative number field self of the given degree,
        or of all possible degrees if degree is 0.  The subfields are returned as
        absolute fields together with an embedding into self.  For the case of the
        field itself, the reverse isomorphism is also provided.

        EXAMPLES::

            sage: PQ.<X> = QQ[]
            sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
            sage: PF.<Y> = F[]
            sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
            sage: K.subfields(2)
            [
            (Number Field in c0 with defining polynomial x^2 - 24*x + 96, Ring morphism:
              From: Number Field in c0 with defining polynomial x^2 - 24*x + 96
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c0 |--> -4*b + 12, None),
            (Number Field in c1 with defining polynomial x^2 - 24*x + 120, Ring morphism:
              From: Number Field in c1 with defining polynomial x^2 - 24*x + 120
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c1 |--> 2*b*a + 12, None),
            (Number Field in c2 with defining polynomial x^2 - 24*x + 72, Ring morphism:
              From: Number Field in c2 with defining polynomial x^2 - 24*x + 72
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c2 |--> -6*a + 12, None)
            ]
            sage: K.subfields(8, 'w')
            [
            (Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9, Ring morphism:
              From: Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9
              To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: w0 |--> (-1/2*b*a + 1/2*b + 1/2)*c, Relative number field morphism:
              From: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              To:   Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9
              Defn: c |--> -1/3*w0^7 + 4*w0^5 - 12*w0^3 + 11*w0
                    a |--> 1/3*w0^6 - 10/3*w0^4 + 5*w0^2
                    b |--> -2/3*w0^6 + 7*w0^4 - 14*w0^2 + 6)
            ]
            sage: K.subfields(3)
            []
        """
        if name is None:
            name = self.variable_name()
        abs = self.absolute_field(name)
        from_abs, to_abs = abs.structure()
        abs_subfields = abs.subfields(degree=degree)
        ans = []
        for K, from_K, to_K in abs_subfields:
            from_K = K.hom([from_abs(from_K(K.gen()))])
            if to_K is not None:
                to_K = RelativeNumberFieldHomomorphism_from_abs(self.Hom(K), to_K*to_abs)
            ans.append((K, from_K, to_K))
        ans = Sequence(ans, immutable=True, cr=ans!=[])
        return ans

    def is_absolute(self):
        r"""
        Returns False, since this is not an absolute field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.is_absolute()
            False
            sage: K.is_relative()
            True
        """
        return False

    def gens(self):
        """
        Return the generators of this relative number field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.gens()
            (a, b)

        TESTS:

        Trivial extensions work like non-trivial ones (:trac:`2220`)::

            sage: NumberField([x^2 - 3, x], 'a').gens()
            (a0, 0)
            sage: NumberField([x, x^2 - 3], 'a').gens()
            (0, a1)

        """
        return ((self._gen_relative(),) +
                tuple(map(self, self.base_field().gens())))

    def _first_ngens(self, n):
        """
        Return the first `n` generators of this relative number field.

        If `n` is greater than the number of generators, the output is
        the same as that of :meth:`gens`.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K._first_ngens(0)
            ()
            sage: K._first_ngens(1)
            (a,)
            sage: K._first_ngens(2)
            (a, b)
            sage: K._first_ngens(3)
            (a, b)
        """
        if n <= 0:
            return ()
        v = (self._gen_relative(),)
        if n > 1:
            v += tuple(map(self, self.base_field()._first_ngens(n - 1)))
        return v

    def ngens(self):
        """
        Return the number of generators of this relative number field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.gens()
            (a, b)
            sage: K.ngens()
            2
        """
        return self.base_field().ngens() + 1

    def gen(self, n=0):
        """
        Return the `n`'th generator of this relative number field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.gens()
            (a, b)
            sage: K.gen(0)
            a
        """
        if n == 0:
            return self._gen_relative()
        return self(self.base_field().gen(n - 1))

    def galois_closure(self, names=None):
        r"""
        Return the absolute number field `K` that is the Galois closure of this
        relative number field.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: K.galois_closure('c')
            Number Field in c with defining polynomial x^16 + 16*x^14 + 28*x^12 + 784*x^10 + 19846*x^8 - 595280*x^6 + 2744476*x^4 + 3212848*x^2 + 29953729
        """
        return self.absolute_field('a').galois_closure(names=names)

    def composite_fields(self, other, names=None, both_maps=False, preserve_embedding=True):
        """
        List of all possible composite number fields formed from self and
        other, together with (optionally) embeddings into the compositum;
        see the documentation for both_maps below.

        Since relative fields do not have ambient embeddings,
        preserve_embedding has no effect.  In every case all possible
        composite number fields are returned.

        INPUT:

        - ``other`` - a number field

        - ``names`` - generator name for composite fields

        - ``both_maps`` - (default: False)  if True, return quadruples
          (F, self_into_F, other_into_F, k) such that self_into_F maps self into
          F, other_into_F maps other into F.  For relative number fields k is
          always None.
        - ``preserve_embedding`` - (default: True) has no effect, but is kept
          for compatibility with the absolute version of this function.  In every
          case the list of all possible compositums is returned.

        OUTPUT:

        -  ``list`` - list of the composite fields, possibly with maps.


        EXAMPLES::

            sage: K.<a, b> = NumberField([x^2 + 5, x^2 - 2])
            sage: L.<c, d> = NumberField([x^2 + 5, x^2 - 3])
            sage: K.composite_fields(L, 'e')
            [Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600]
            sage: K.composite_fields(L, 'e', both_maps=True)
            [[Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600,
              Relative number field morphism:
              From: Number Field in a with defining polynomial x^2 + 5 over its base field
             To:   Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600
              Defn: a |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e
                    b |--> -21/166400*e^7 + 73/20800*e^5 - 779/10400*e^3 + 7/260*e,
              Relative number field morphism:
              From: Number Field in c with defining polynomial x^2 + 5 over its base field
              To:   Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600
              Defn: c |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e
                    d |--> -3/25600*e^7 + 7/1600*e^5 - 147/1600*e^3 + 1/40*e,
              None]]
        """
        if not isinstance(other, NumberField_generic):
            raise TypeError("other must be a number field.")
        if names is None:
            sv = self.variable_name()
            ov = other.variable_name()
            names = sv + (ov if ov != sv else "")

        self_abs = self.absolute_field('w')
        abs_composites = self_abs.composite_fields(other, names=names, both_maps=both_maps)

        m = self.absolute_degree()

        if not both_maps:
            rets = []
            for F in abs_composites:
                if F.absolute_degree() == m:
                    F = self
                rets.append(F)
            return rets

        from_self_abs, to_self_abs = self_abs.structure()

        rets = []
        for F, self_abs_to_F, other_to_F, k in abs_composites:
            self_to_F = RelativeNumberFieldHomomorphism_from_abs(self.Hom(F), self_abs_to_F*to_self_abs)
            if F.absolute_degree() == m:
                if other.is_absolute():
                    other_to_F = other.hom([(from_self_abs*(~self_abs_to_F)*other_to_F)(other.gen())])
                else:
                    other_to_F = RelativeNumberFieldHomomorphism_from_abs(self.Hom(self), from_self_abs*(~self_abs_to_F)*other_to_F)
                self_to_F = RelativeNumberFieldHomomorphism_from_abs(self.Hom(self), from_self_abs)
                F = self
            rets.append([F, self_to_F, other_to_F, None])
        return rets

    def absolute_degree(self):
        """
        The degree of this relative number field over the rational field.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.absolute_degree()
            6
        """
        return self.absolute_polynomial().degree()

    def relative_degree(self):
        r"""
        Returns the relative degree of this relative number field.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.relative_degree()
            2
        """
        return self.relative_polynomial().degree()

    def degree(self):
        """
        The degree, unqualified, of a relative number field is deliberately
        not implemented, so that a user cannot mistake the absolute degree
        for the relative degree, or vice versa.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.degree()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field you must use relative_degree or absolute_degree as appropriate
        """
        raise NotImplementedError("For a relative number field you must use relative_degree or absolute_degree as appropriate")

    @cached_method
    def _maximal_order(self, v=(), assume_maximal='non-maximal-non-unique'):
        """
        Implements :meth:`NumberField_generic.maximal_order` for relative
        number fields.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.maximal_order() is K.maximal_order()  # indirect doctest
            True

        """
        absolute_order = self.absolute_field('z').maximal_order(v=v, assume_maximal=assume_maximal)

        return RelativeOrder(self, absolute_order, is_maximal=assume_maximal, is_maximal_at=v)

    def _repr_(self):
        """
        Return string representation of this relative number field.

        The base field is not part of the string representation.  To
        find out what the base field is use :meth:`~base_field`.

        EXAMPLES::

            sage: k.<a, b> = NumberField([x^5 + 2, x^7 + 3])
            sage: repr(k) # indirect doctest
            'Number Field in a with defining polynomial x^5 + 2 over its base field'
            sage: k.base_field()
            Number Field in b with defining polynomial x^7 + 3
        """

        return "Number Field in %s with defining polynomial %s over its base field"%(self.variable_name(), self.relative_polynomial())

    def _Hom_(self, codomain, category=None):
        """
        Return homset of homomorphisms from this relative number field
        to the codomain.

        EXAMPLES:

        This function is implicitly called by the ``Hom`` method or
        function::

            sage: K.<a,b> = NumberField([x^3 - 2, x^2+1])
            sage: K.Hom(K) # indirect doctest
            Automorphism group of Number Field in a with defining polynomial x^3 - 2 over its base field
            sage: type(K.Hom(K))
            <class 'sage.rings.number_field.homset.RelativeNumberFieldHomset_with_category'>

        TESTS::

            sage: H = End(K)
            sage: loads(dumps(H)) is H
            True
        """
        if not is_NumberFieldHomsetCodomain(codomain):
            raise TypeError("{} is not suitable as codomain for homomorphisms from {}".format(codomain, self))

        from sage.rings.number_field.homset import RelativeNumberFieldHomset
        return RelativeNumberFieldHomset(self, codomain, category)

    def _latex_(self):
        r"""
        Return a `\LaTeX` representation of the extension.

        EXAMPLES::

            sage: x = polygen(QQ)
            sage: K.<a> = NumberField(x^3 - 2)
            sage: t = polygen(K)
            sage: K.extension(t^2+t+a, 'b')._latex_()
            '( \\Bold{Q}[a]/(a^{3} - 2) )[b]/(b^{2} + b + a)'
        """
        latex_name = self.latex_variable_names()[0]
        return "( %s )[%s]/(%s)"%(latex(self.base_field()), latex_name,
                              self.relative_polynomial()._latex_(latex_name))

    def _coerce_from_other_number_field(self, x):
        """
        Coerce a number field element x into this number field.

        In most cases this currently doesn't work (since it is
        barely implemented) -- it only works for constants.

        INPUT:

        - ``x`` -- an element of some number field

        EXAMPLES::

            sage: K.<a> = NumberField(x^3 + 2)
            sage: L.<b> = NumberField(x^2 + 1)
            sage: K._coerce_from_other_number_field(L(2/3))
            2/3
        """
        if x.parent() is self.base_ring():
            return self.__base_inclusion(x)

        f = x.polynomial()
        if f.degree() <= 0:
            return self._element_class(self, f[0])
        # todo: more general coercion if embedding have been asserted
        raise TypeError("Cannot coerce element into this number field")

    def _convert_non_number_field_element(self, x):
        r"""
        Convert the non-number field element `x` into this number field.

        INPUT:

        - ``x`` -- a non number field element, e.g., a list,
          integer, rational, or polynomial.

        EXAMPLES::

            sage: x = polygen(ZZ)
            sage: K.<a> = NumberField(x^5 + 2)
            sage: L.<b> = K.extension(x^2 + 3*a)
            sage: R.<u> = QQ[]
            sage: S.<t> = R[]

            sage: L(a + b)
            b + a

            sage: L(5*t*(1 + u) + 2/3*u)
            (5*a + 5)*b + 2/3*a
            sage: L(0*t + 2/3)
            2/3
            sage: L(1/2*t + 5)
            1/2*b + 5

        This seems reasonable::

            sage: L(t*5)
            5*b

        This is misleading, but correct!  It is more often desired
        to make a number field element given by rational
        coefficients of the relative power basis (so 2*b^2 + 3)
        than it is to create the constant term of such an element,
        which is what would happen if L(u*5) gave 5*a.::

            sage: L(u*5)
            5*b

            sage: L([1, 1/2])
            1/2*b + 1
            sage: L([ a, 1/2 + a/3 ])
            (1/3*a + 1/2)*b + a

            sage: L([ 1 ])
            Traceback (most recent call last):
            ...
            ValueError: Length must be equal to the degree of this number field

        TESTS:

        Examples from :trac:`4727`::

            sage: K.<j,b> = QQ[sqrt(-1), sqrt(2)]
            sage: j
            I
            sage: j.list()
            [0, 1]
            sage: K(j.list())
            I
            sage: (b*j + 1/2).list()
            [1/2, sqrt2]
            sage: K((b*j + 1/2).list())
            sqrt2*I + 1/2

        Examples from :trac:`4869`::

            sage: K.<z> = CyclotomicField(7)
            sage: Ky.<y> = PolynomialRing(K)
            sage: L.<a> = K.extension(y^2 + 1)
            sage: K(K.polynomial_ring().random_element()) # random
            -12*z^2 + 1/2*z - 1/95
            sage: L(L.polynomial_ring().random_element()) # random
            (z^5 + 1/3*z^4 - z^3 + z^2 - z + 2/3)*a + 1/4*z^5 - 7/2*z^4 + 5/3*z^3 - 1/4*z^2 + 3/2*z - 1

        Examples from :trac:`11307`::

            sage: L = NumberField([x^2 + 1, x^2 - 3], 'a')
            sage: L(L)
            Traceback (most recent call last):
            ...
            TypeError: unable to convert Number Field in a0 with defining polynomial x^2 + 1 over its base field to Number Field in a0 with defining polynomial x^2 + 1 over its base field
            sage: L in L
            False

        We construct the composite of three quadratic fields, then
        coerce from the quartic subfield of the relative extension::

            sage: k.<a,b,c> = NumberField([x^2 + 5, x^2 + 3, x^2 + 1])
            sage: m = k.base_field(); m
            Number Field in b with defining polynomial x^2 + 3 over its base field
            sage: k(m.0)
            b
            sage: k(2/3)
            2/3
            sage: k(m.0^4)
            9

            sage: x = polygen(ZZ)
            sage: K.<a> = NumberField(x^2 + 2, 'a')
            sage: L.<b> = K.extension(x - a, 'b')
            sage: L(a)
            a
            sage: L(b+a)
            2*a
            sage: K.<a> = NumberField(x^5 + 2, 'a')
            sage: L.<b> = K.extension(x - a, 'b')
            sage: L(a)
            a
            sage: L(a**3)
            a^3
            sage: L(a**2+b)
            a^2 + a
            sage: L.<b> = K.extension(x + a/2, 'b')
            sage: L(a)
            a
            sage: L(a).polynomial()
            -x
            sage: L(a).minpoly()
            x - a
            sage: L(a).absolute_minpoly()
            x^5 + 2
            sage: L(b)
            -1/2*a
            sage: L(b).polynomial()
            1/2*x
            sage: L(b).absolute_minpoly()
            x^5 - 1/16
            sage: L(b).minpoly()
            x + 1/2*a

        ::

            sage: K.<a> = NumberField(x^5+2)
            sage: R.<y> = K[]
            sage: L.<x0> = K.extension(y + a**2)
            sage: L(a)
            a
        """
        if isinstance(x, polynomial_element.Polynomial):
            # we have been given a polynomial, change it to an absolute polynomial
            K = self.base_ring()
            R = self.polynomial_ring()
            if QQ.has_coerce_map_from(x.parent().base_ring()):
                # special case absolute polynomials -- they should be
                # in terms of the relative generator
                x = R(x.list())
            # this should work for base_ring()['x'] and QQ['base']['ext']
            x = self.polynomial_ring()(x)
            f = R( [ K(coeff) for coeff in x.list() ] )
            return self._element_class(self, f(self.gen()).polynomial() )

        # Anything else: use the code for generic number fields
        return super(NumberField_relative, self)._convert_non_number_field_element(x)

    def _coerce_map_from_(self, R):
        """
        Canonical coercion of x into this relative number field.

        Currently integers, rationals, the base field, and this field
        itself coerce canonically into this field (and hence so does
        anything that coerces into one of these).

        EXAMPLES::

            sage: k.<a> = NumberField([x^5 + 2, x^7 + 3])
            sage: b = k(k.base_field().gen())
            sage: b = k.coerce(k.base_field().gen()) # indirect doctest
            sage: b^7
            -3
            sage: k.coerce(2/3)
            2/3
            sage: c = a + b # no output
        """
        if R is int:
            return self._generic_coerce_map(R)
        elif R in (ZZ, QQ, self.base_field()):
            return self._generic_coerce_map(R)
        if is_NumberFieldOrder(R) and R.number_field() is self:
            return self._generic_coerce_map(R)
        mor = self.base_field()._internal_coerce_map_from(R)
        if mor is not None:
            return self._internal_coerce_map_from(self.base_field()) * mor

    def _element_constructor_(self, x, check=True):
        """
        Construct a relative number field element from ``x``.

        EXAMPLES:

        We can create relative number field elements from PARI::

            sage: y = polygen(QQ)
            sage: K.<a> = NumberField(y^2 + y + 1)
            sage: x = polygen(K)
            sage: L.<b> = NumberField(x^4 + a*x + 2)
            sage: e = a.__pari__(); e
            Mod(y, y^2 + y + 1)
            sage: L(e)  # Conversion from PARI base field element
            a
            sage: e = (a*b).__pari__('x'); e
            Mod(-x^4 - 2, x^8 - x^5 + 4*x^4 + x^2 - 2*x + 4)
            sage: L(e)  # Conversion from PARI absolute number field element
            a*b
            sage: e = L.pari_rnf().rnfeltabstorel(e); e
            Mod(Mod(y, y^2 + y + 1)*x, x^4 + Mod(y, y^2 + y + 1)*x + 2)
            sage: L(e)  # Conversion from PARI relative number field element
            a*b
            sage: e = pari('Mod(0, x^8 + 1)'); L(e)  # Wrong modulus
            Traceback (most recent call last):
            ...
            TypeError: cannot convert PARI element Mod(0, x^8 + 1) into Number Field in b with defining polynomial x^4 + a*x + 2 over its base field

        We test a relative number field element created "by hand"::

            sage: e = pari("Mod(Mod(y, y^2 + y + 1)*x^2 + Mod(1, y^2 + y + 1), x^4 + y*x + 2)")
            sage: L(e)
            a*b^2 + 1

        A wrong modulus yields an error::

            sage: e = pari('Mod(y*x, x^4 + y^2*x + 2)'); L(e)
            Traceback (most recent call last):
            ...
            TypeError: cannot convert PARI element Mod(y*x, x^4 + y^2*x + 2) into Number Field in b with defining polynomial x^4 + a*x + 2 over its base field
        """
        # If x is a *relative* PARI number field element, convert it
        # to an absolute element.
        if isinstance(x, pari_gen) and x.type() == "t_POLMOD":
            modulus = x.mod()
            if (modulus == self.pari_relative_polynomial()
                or modulus == self.pari_absolute_base_polynomial()):
                x = self._pari_rnfeq()._eltreltoabs(x.liftpol())
                check = False
        return NumberField_generic._element_constructor_(self, x, check=check)

    def __base_inclusion(self, element):
        """
        Given an element of the base field, give its inclusion into
        this extension in terms of the generator of this field.

        This is called by the canonical coercion map on elements from
        the base field.

        EXAMPLES::

            sage: k.<a> = NumberField([x^2 + 3, x^2 + 1])
            sage: m = k.base_field(); m
            Number Field in a1 with defining polynomial x^2 + 1
            sage: k.coerce(m.0 + 2/3) # indirect doctest
            a1 + 2/3
            sage: s = k.coerce(m.0); s
            a1
            sage: s^2
            -1

        This implicitly tests this coercion map::

            sage: K.<a> = NumberField([x^2 + p for p in [5,3,2]])
            sage: K.coerce(K.base_field().0)
            a1
            sage: K.coerce(K.base_field().0)^2
            -3

        TESTS:

        Check that :trac:`5828` is solved::

            sage: K.<w> = QuadraticField(-1)
            sage: KX.<X> = K[]
            sage: H.<h> = K.extension(X-1)
            sage: H(w)
            w
        """
        abs_base, from_abs_base, to_abs_base = self.absolute_base_field()
        # Write element in terms of the absolute base field
        element = self.base_field().coerce(element)
        element = to_abs_base(element)
        # Express element as a polynomial in the absolute generator of self
        nfzk = self._pari_nfzk()
        expr_x = self._pari_base_nf()._nfeltup(element._pari_polynomial(), nfzk)
        # We do NOT call self(...) because this code is called by
        # __init__ before we initialize self.gens(), and self(...)
        # uses self.gens()
        return self._element_constructor_(expr_x, check=False)

    def _fractional_ideal_class_(self):
        """
        Return the Python class used to represent ideals of a relative
        number field.

        EXAMPLES::

            sage: k.<a> = NumberField([x^5 + 2, x^7 + 3])
            sage: k._fractional_ideal_class_ ()
            <class 'sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel'>
        """
        return sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel

    def _pari_base_bnf(self, proof=False, units=True):
        r"""
        Return the PARI bnf (big number field) representation of the
        absolute base field in terms of the pari variable ``y``, suitable
        for extension by the pari variable ``x``.

        All caching is done by the absolute base field.

        INPUT:

        - ``proof`` (bool, default True) -- if True, certify
          correctness of calculations (not assuming GRH).

        EXAMPLES::

            sage: k.<a> = NumberField([x^3 + 2, x^2 + 2])
            sage: k._pari_base_bnf()
            [[;], matrix(0,3), [;], ...]
        """
        abs_base, from_abs_base, to_abs_base = self.absolute_base_field()
        return abs_base.pari_bnf(proof, units)

    def _pari_base_nf(self):
        r"""
        Return the PARI number field representation of the absolute
        base field, in terms of the pari variable ``y``, suitable for
        extension by the pari variable ``x``.

        All caching is done by the absolute base field.

        EXAMPLES::

            sage: y = polygen(QQ,'y')
            sage: k.<a> = NumberField([y^3 + 2, y^2 + 2])
            sage: k._pari_base_nf()
            [y^2 + 2, [0, 1], -8, 1, ..., [1, 0, 0, -2; 0, 1, 1, 0]]
        """
        abs_base, from_abs_base, to_abs_base = self.absolute_base_field()
        return abs_base.pari_nf()

    def is_galois(self):
        r"""
        For a relative number field, ``is_galois()`` is deliberately not
        implemented, since it is not clear whether this would mean "Galois over
        `\QQ`" or "Galois over the given base field". Use either ``is_galois_absolute()`` or ``is_galois_relative()`` respectively.

        EXAMPLES::

            sage: k.<a> =NumberField([x^3 - 2, x^2 + x + 1])
            sage: k.is_galois()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field L you must use either L.is_galois_relative() or L.is_galois_absolute() as appropriate
        """
        raise NotImplementedError("For a relative number field L you must use either L.is_galois_relative() or L.is_galois_absolute() as appropriate")

    def is_galois_relative(self):
        r"""
        Return True if for this relative extension `L/K`, `L` is a
        Galois extension of `K`.

        EXAMPLES::

            sage: K.<a> = NumberField(x^3 - 2)
            sage: y = polygen(K)
            sage: L.<b> = K.extension(y^2 - a)
            sage: L.is_galois_relative()
            True
            sage: M.<c> = K.extension(y^3 - a)
            sage: M.is_galois_relative()
            False

        The next example previously gave a wrong result; see :trac:`9390`::

            sage: F.<a, b> = NumberField([x^2 - 2, x^2 - 3])
            sage: F.is_galois_relative()
            True
        """
        d = self.relative_degree()
        if d <= 2:
            return True
        else:
            rel_poly = self.relative_polynomial()
            return d == len(rel_poly.base_extend(self).factor())

    def is_galois_absolute(self):
        r"""
        Return True if for this relative extension `L/K`, `L` is a Galois extension of `\QQ`.

        EXAMPLES::

            sage: K.<a> = NumberField(x^3 - 2)
            sage: y = polygen(K); L.<b> = K.extension(y^2 - a)
            sage: L.is_galois_absolute()
            False

        """
        f = self.absolute_polynomial()
        return f.galois_group(pari_group=True).order() == self.absolute_degree()

    def is_isomorphic_relative(self, other, base_isom=None):
        r"""
        For this relative extension `L/K` and another relative extension `M/K`, return True
        if there is a `K`-linear isomorphism from `L` to `M`. More generally, ``other`` can be a
        relative extension `M/K^\prime` with ``base_isom`` an isomorphism from `K` to
        `K^\prime`.

        EXAMPLES::

            sage: K.<z9> = NumberField(x^6 + x^3 + 1)
            sage: R.<z> = PolynomialRing(K)
            sage: m1 = 3*z9^4 - 4*z9^3 - 4*z9^2 + 3*z9 - 8
            sage: L1 = K.extension(z^2 - m1, 'b1')
            sage: G = K.galois_group(); gamma = G.gen()
            sage: m2 = (gamma^2)(m1)
            sage: L2 = K.extension(z^2 - m2, 'b2')
            sage: L1.is_isomorphic_relative(L2)
            False
            sage: L1.is_isomorphic(L2)
            True
            sage: L3 = K.extension(z^4 - m1, 'b3')
            sage: L1.is_isomorphic_relative(L3)
            False

        If we have two extensions over different, but isomorphic, bases, we can compare them by
        letting ``base_isom`` be an isomorphism from self's base field to other's base field::

            sage: Kcyc.<zeta9> = CyclotomicField(9)
            sage: Rcyc.<zcyc> = PolynomialRing(Kcyc)
            sage: phi1 = K.hom([zeta9])
            sage: m1cyc = phi1(m1)
            sage: L1cyc = Kcyc.extension(zcyc^2 - m1cyc, 'b1cyc')
            sage: L1.is_isomorphic_relative(L1cyc, base_isom=phi1)
            True
            sage: L2.is_isomorphic_relative(L1cyc, base_isom=phi1)
            False
            sage: phi2 = K.hom([phi1((gamma^(-2))(z9))])
            sage: L1.is_isomorphic_relative(L1cyc, base_isom=phi2)
            False
            sage: L2.is_isomorphic_relative(L1cyc, base_isom=phi2)
            True

        Omitting ``base_isom`` raises a ValueError when the base fields are not identical::

            sage: L1.is_isomorphic_relative(L1cyc)
            Traceback (most recent call last):
            ...
            ValueError: other does not have the same base field as self, so an isomorphism from self's base_field to other's base_field must be provided using the base_isom parameter.

        The parameter ``base_isom`` can also be used to check if the relative extensions are
        Galois conjugate::

            sage: for g in G:
            ....:   if L1.is_isomorphic_relative(L2, g.as_hom()):
            ....:       print(g.as_hom())
            Ring endomorphism of Number Field in z9 with defining polynomial x^6 + x^3 + 1
              Defn: z9 |--> z9^4
        """
        if is_RelativeNumberField(other):
            s_base_field = self.base_field()
            o_base_field = other.base_field()
            if base_isom is None:
                if s_base_field is o_base_field:
                    return self.relative_degree() == other.relative_degree() and len(self.relative_polynomial().roots(other)) > 0
                raise ValueError("other does not have the same base field as self, so an isomorphism from self's base_field to other's base_field must be provided using the base_isom parameter.")
            if s_base_field.absolute_degree() != o_base_field.absolute_degree():
                raise ValueError("The base fields are not isomorphic.")
            if base_isom.domain() is s_base_field and base_isom.codomain() is o_base_field:
                if s_base_field.absolute_degree() != o_base_field.absolute_degree():
                    raise ValueError("The base fields are not isomorphic.")
                if not self.relative_degree() == other.relative_degree():
                    return False
                R = PolynomialRing(o_base_field, 'x')
                F = R([base_isom(_) for _ in self.relative_polynomial()])
                return len(F.roots(other)) > 0
            raise ValueError("base_isom is not a homomorphism from self's base_field to other's base_field")
        raise ValueError("other must be a relative number field.")

    def is_CM_extension(self):
        """
        Return True is this is a CM extension, i.e. a totally imaginary
        quadratic extension of a totally real field.

        EXAMPLES::

            sage: F.<a> = NumberField(x^2 - 5)
            sage: K.<z> = F.extension(x^2 + 7)
            sage: K.is_CM_extension()
            True
            sage: K = CyclotomicField(7)
            sage: K_rel = K.relativize(K.gen() + K.gen()^(-1), 'z')
            sage: K_rel.is_CM_extension()
            True
            sage: F = CyclotomicField(3)
            sage: K.<z> = F.extension(x^3 - 2)
            sage: K.is_CM_extension()
            False

        A CM field K such that K/F is not a CM extension

        ::

            sage: F.<a> = NumberField(x^2 + 1)
            sage: K.<z> = F.extension(x^2 - 3)
            sage: K.is_CM_extension()
            False
            sage: K.is_CM()
            True

        """

        try:
            return self.__is_CM_extension
        except(AttributeError):
            pass

        if self.relative_degree() == 2:
            if self.base_field().is_totally_real():
                if self.is_totally_imaginary():
                    self.__is_CM_extension = True
                    self.__is_CM = True
                    self.__max_tot_real_sub = [self.base_field(), self._internal_coerce_map_from(self.base_field())]
                    return True
        self.__is_CM_extension = False
        return False

    @cached_method(key=lambda self, base, basis, map: (base or self.base_ring(), basis, map))
    def free_module(self, base=None, basis=None, map=True):
        """
        Return a vector space over a specified subfield that is isomorphic to this number field,
        together with the isomorphisms in each direction.

        INPUT:

        - ``base`` -- a subfield

        - ``basis`` -- (optional) a list of elements giving a basis over the subfield

        - ``map`` -- (default ``True``) whether to return isomorphisms to and from the vector space

        EXAMPLES::

            sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K
            Number Field in a with defining polynomial x^2 + 2 over its base field
            sage: V, from_V, to_V = K.free_module()
            sage: to_V(K.0)
            (0, 1)
            sage: W, from_W, to_W = K.free_module(base=QQ)
            sage: w = to_W(K.0); len(w)
            18
            sage: w[0]
            -127917622658689792301282/48787705559800061938765
        """
        if basis is not None:
            raise NotImplementedError
        if base is None:
            base = self.base_field()
        if base is self.base_field():
            V = self.base_field()**self.relative_degree()
            if not map:
                return V
            fr = maps.MapRelativeVectorSpaceToRelativeNumberField(V, self)
            to = maps.MapRelativeNumberFieldToRelativeVectorSpace(self, V)
        elif base is QQ:
            if not map:
                return QQ**self.absolute_degree()
            K = self.absolute_field('a')
            from_K, to_K = K.structure()
            V, from_V, to_V = K.free_module()
            fr = maps.MapVectorSpaceToRelativeNumberField(V, self, from_V, from_K)
            to = maps.MapRelativeNumberFieldToVectorSpace(self, V, to_K, to_V)
        else:
            raise NotImplementedError
        return V, fr, to

    def relative_vector_space(self, base=None, *args, **kwds):
        """
        Return vector space over the base field of self and isomorphisms
        from the vector space to self and in the other direction.

        EXAMPLES::

            sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K
            Number Field in a with defining polynomial x^2 + 2 over its base field
            sage: V, from_V, to_V = K.relative_vector_space()
            sage: from_V(V.0)
            1
            sage: to_V(K.0)
            (0, 1)
            sage: from_V(to_V(K.0))
            a
            sage: to_V(from_V(V.0))
            (1, 0)
            sage: to_V(from_V(V.1))
            (0, 1)

        The underlying vector space and maps is cached::

            sage: W, from_V, to_V = K.relative_vector_space()
            sage: V is W
            True
        """
        if base is not None and base is not self.base_field():
            raise ValueError("Relative vector space base must be the base field")
        return self.free_module(self.base_field(), *args, **kwds)

    def absolute_vector_space(self, base=None, *args, **kwds):
        r"""
        Return vector space over `\QQ` of ``self`` and isomorphisms from
        the vector space to ``self`` and in the other direction.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^3 + 3, x^3 + 2]); K
            Number Field in a with defining polynomial x^3 + 3 over its base field
            sage: V,from_V,to_V = K.absolute_vector_space(); V
            Vector space of dimension 9 over Rational Field
            sage: from_V
            Isomorphism map:
              From: Vector space of dimension 9 over Rational Field
              To:   Number Field in a with defining polynomial x^3 + 3 over its base field
            sage: to_V
            Isomorphism map:
              From: Number Field in a with defining polynomial x^3 + 3 over its base field
              To:   Vector space of dimension 9 over Rational Field
            sage: c = (a+1)^5; c
            7*a^2 - 10*a - 29
            sage: to_V(c)
            (-29, -712/9, 19712/45, 0, -14/9, 364/45, 0, -4/9, 119/45)
            sage: from_V(to_V(c))
            7*a^2 - 10*a - 29
            sage: from_V(3*to_V(b))
            3*b
        """
        if base is not None and base is not QQ:
            raise ValueError("Absolute vector space base must be QQ")
        return self.free_module(QQ, *args, **kwds)

    def vector_space(self, *args, **kwds):
        r"""
        For a relative number field, ``vector_space()`` is
        deliberately not implemented, so that a user cannot confuse
        :meth:`~relative_vector_space` with :meth:`~absolute_vector_space`.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
            sage: K.vector_space()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field L you must use either L.relative_vector_space() or L.absolute_vector_space() as appropriate

        """
        raise NotImplementedError("For a relative number field L you must use either L.relative_vector_space() or L.absolute_vector_space() as appropriate")

    def absolute_base_field(self):
        r"""
        Return the base field of this relative extension, but viewed
        as an absolute field over `\QQ`.

        EXAMPLES::

            sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 3, x^3 + 2])
            sage: K
            Number Field in a with defining polynomial x^2 + 2 over its base field
            sage: K.base_field()
            Number Field in b with defining polynomial x^3 + 3 over its base field
            sage: K.absolute_base_field()[0]
            Number Field in a0 with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1
            sage: K.base_field().absolute_field('z')
            Number Field in z with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1
        """
        return self.__absolute_base_field

    @cached_method
    def _pari_rnfeq(self):
        """
        Return PARI data attached to this relative number field.

        OUTPUT:

        A 5-element PARI vector containing an absolute polynomial and
        further data needed for ``eltabstorel`` and ``eltreltoabs``,
        obtained from PARI's ``nf_rnfeq`` function.  This means that
        we do not have to initialize a full PARI ``rnf`` structure.

        TESTS::

            sage: K.<a> = NumberField(x^2 + 2)
            sage: x = polygen(K)
            sage: L.<b> = K.extension(x^5 + 2*a)
            sage: L._pari_rnfeq()
            [x^10 + 8, -1/2*x^5, 0, y^2 + 2, x^5 + 2*y]
            sage: x = polygen(ZZ)
            sage: NumberField(x^10 + 8, 'a').is_isomorphic(L)
            True

        Initialization is lazy enough to allow arithmetic in massive fields::

            sage: K.<a> = NumberField(x^10 + 2000*x + 100001)
            sage: x = polygen(K)
            sage: L.<b> = K.extension(x^10 + 2*a)
            sage: L._pari_rnfeq()
            [x^100 - 1024000*x^10 + 102401024, -1/2*x^10, 0, y^10 + 2000*y + 100001, x^10 + 2*y]
            sage: a + b
            b + a
            sage: b^100
            -2048000*a - 102401024
            sage: (-2*a)^10
            -2048000*a - 102401024
        """
        f = self.pari_absolute_base_polynomial()
        g = self.pari_relative_polynomial()
        return f._nf_rnfeq(g)

    @cached_method
    def _pari_nfzk(self):
        """
        Return PARI data needed for constructing relative number field
        elements from elements of the base field.

        TESTS::

            sage: K.<a> = NumberField(x^2 - 2)
            sage: L.<b> = K.extension(x^2 - 3)
            sage: L._pari_nfzk()
            [2, -x^3 + 9*x]
        """
        return self._pari_base_nf()._nf_nfzk(self._pari_rnfeq())

    @cached_method
    def _pari_relative_structure(self):
        r"""
        Return data relating the Sage and PARI relative polynomials.

        OUTPUT:

        Let `L` be this relative number field, let `K` be its base
        field, and let `f` be the defining polynomial of `L` over `K`.
        This method returns a triple ``(g, alpha, beta)``, where

        - ``g`` is the defining relative polynomial of the PARI
          ``rnf`` structure (see :meth:`pari_rnf`);

        - ``alpha`` is the image of `x \bmod f` under some isomorphism
          `\phi\colon K[x]/(f) \to K[x]/(g)`;

        - ``beta`` is the image of `x \bmod g` under the inverse
          isomorphism `\phi^{-1}\colon K[x]/(g) \to K[x]/(f)`.

        EXAMPLES::

        If the defining polynomials are monic and integral, the result
        satisfies ``g = f`` and ``alpha = beta = x``::

            sage: R.<x> = QQ[]
            sage: K.<a> = NumberField(x^2 + 1)
            sage: L.<b> = K.extension(x^2 - a)
            sage: L._pari_relative_structure()
            (Mod(1, y^2 + 1)*x^2 + Mod(-y, y^2 + 1),
             Mod(x, Mod(1, y^2 + 1)*x^2 + Mod(-y, y^2 + 1)),
             Mod(x, Mod(1, y^2 + 1)*x^2 + Mod(-y, y^2 + 1)))

        An example where the base field is defined by a monic integral
        polynomial, but the extension is not::

            sage: K.<a> = NumberField(x^2 + 1)
            sage: L.<b> = K.extension(x^2 - 1/2)
            sage: L._pari_relative_structure()
            (x^2 + Mod(-y, y^2 + 1),
             Mod(Mod(1/2*y - 1/2, y^2 + 1)*x, x^2 + Mod(-y, y^2 + 1)),
             Mod(Mod(-y - 1, y^2 + 1)*x, Mod(1, y^2 + 1)*x^2 + Mod(-1/2, y^2 + 1)))

        An example where both fields are defined by non-integral or
        non-monic polynomials::

            sage: K.<a> = NumberField(2*x^2 + 1)
            sage: L.<b> = K.extension(x^2 - 1/3)
            sage: L._pari_relative_structure()
            (x^2 + Mod(y, y^2 + 2)*x + 1,
             Mod(Mod(-1/3*y, y^2 + 2)*x + Mod(1/3, y^2 + 2), x^2 + Mod(y, y^2 + 2)*x + 1),
             Mod(Mod(3/2*y, y^2 + 2)*x + Mod(-1/2*y, y^2 + 2), Mod(1, y^2 + 2)*x^2 + Mod(-1/3, y^2 + 2)))

        Note that in the last example, the *absolute* defining
        polynomials is the same for Sage and PARI, even though this is
        not the case for the base field::

            sage: K._pari_absolute_structure()
            (y^2 + 2, Mod(1/2*y, y^2 + 2), Mod(2*y, y^2 + 1/2))
            sage: L._pari_absolute_structure()
            (y^4 + 4*y^2 + 1, Mod(y, y^4 + 4*y^2 + 1), Mod(y, y^4 + 4*y^2 + 1))
        """
        f = self.relative_polynomial()._pari_with_name('x')
        if f.pollead() == f.content().lift().content().denominator() == 1:
            g = f
            alpha = beta = f.variable().Mod(f)
        elif f.poldegree() == 1:
            # PARI's rnfpolredbest() does not always return a
            # polynomial with integral coefficients in this case.
            from sage.libs.pari.all import pari
            g = f.variable()
            alpha = -f[0]/f[1]
            beta = pari(0).Mod(f)
        else:
            g, alpha = self._pari_base_nf().rnfpolredbest(f, flag=1)
            beta = alpha.modreverse()
        return g, alpha, beta

    @cached_method
    def _gen_relative(self):
        r"""
        Return root of defining polynomial, which is a generator of
        the relative number field over the base.

        EXAMPLES::

            sage: k.<a> = NumberField(x^2+1); k
            Number Field in a with defining polynomial x^2 + 1
            sage: y = polygen(k)
            sage: m.<b> = k.extension(y^2+3); m
            Number Field in b with defining polynomial x^2 + 3 over its base field
            sage: c = m.gen(); c # indirect doctest
            b
            sage: c^2 + 3
            0

        An example where the defining polynomials are not monic or
        integral::

            sage: K.<a> = NumberField(3*x^2 + 1)
            sage: L.<b> = K.extension(x^2 - 1/2)
            sage: L._gen_relative()
            b
        """
        alpha = self._pari_relative_structure()[1].liftpol()
        return self._element_constructor_(self._pari_rnfeq()._eltreltoabs(alpha), check=False)

    @cached_method
    def pari_rnf(self):
        r"""
        Return the PARI relative number field object associated
        to this relative extension.

        EXAMPLES::

            sage: k.<a> = NumberField([x^4 + 3, x^2 + 2])
            sage: k.pari_rnf()
            [x^4 + 3, [364, -10*x^7 - 87*x^5 - 370*x^3 - 41*x], [108, 3], ...]
        """
        return self._pari_base_nf().rnfinit(self.pari_relative_polynomial())

    def pari_absolute_base_polynomial(self):
        r"""
        Return the PARI polynomial defining the absolute base field, in ``y``.

        EXAMPLES::

            sage: x = polygen(ZZ)
            sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 3]); K
            Number Field in a with defining polynomial x^2 + 2 over its base field
            sage: K.pari_absolute_base_polynomial()
            y^2 + 3
            sage: type(K.pari_absolute_base_polynomial())
            <class 'cypari2.gen.Gen'>
            sage: z = ZZ['z'].0
            sage: K.<a, b, c> = NumberField([z^2 + 2, z^2 + 3, z^2 + 5]); K
            Number Field in a with defining polynomial z^2 + 2 over its base field
            sage: K.pari_absolute_base_polynomial()
            y^4 + 16*y^2 + 4
            sage: K.base_field()
            Number Field in b with defining polynomial z^2 + 3 over its base field
            sage: len(QQ['y'](K.pari_absolute_base_polynomial()).roots(K.base_field()))
            4
            sage: type(K.pari_absolute_base_polynomial())
            <class 'cypari2.gen.Gen'>
        """
        abs_base, from_abs_base, to_abs_base = self.absolute_base_field()
        return abs_base.pari_polynomial('y')

    def pari_relative_polynomial(self):
        r"""
        Return the PARI relative polynomial associated to this number
        field.

        This is always a polynomial in x and y, suitable for PARI's
        rnfinit function.  Notice that if this is a relative extension
        of a relative extension, the base field is the absolute base
        field.

        EXAMPLES::

            sage: k.<i> = NumberField(x^2 + 1)
            sage: m.<z> = k.extension(k['w']([i,0,1]))
            sage: m
            Number Field in z with defining polynomial w^2 + i over its base field
            sage: m.pari_relative_polynomial()
            Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1)

            sage: l.<t> = m.extension(m['t'].0^2 + z)
            sage: l.pari_relative_polynomial()
            Mod(1, y^4 + 1)*x^2 + Mod(y, y^4 + 1)
        """
        return self._pari_relative_structure()[0]

    def number_of_roots_of_unity(self):
        """
        Return number of roots of unity in this relative field.

        EXAMPLES::

            sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] )
            sage: K.number_of_roots_of_unity()
            24
        """
        return self.absolute_field('a').number_of_roots_of_unity()

    def roots_of_unity(self):
        """
        Return all the roots of unity in this relative field, primitive or not.

        EXAMPLES::

            sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] )
            sage: rts = K.roots_of_unity()
            sage: len(rts)
            24
            sage: all(u in rts for u in [b*a, -b^2*a - b^2, b^3, -a, b*a + b])
            True
        """
        abs = self.absolute_field('a')
        from_abs, _ = abs.structure()
        return [from_abs(x) for x in abs.roots_of_unity()]

    def absolute_generator(self):
        r"""
        Return the chosen generator over `\QQ` for this relative number field.

        EXAMPLES::

            sage: y = polygen(QQ,'y')
            sage: k.<a> = NumberField([y^2 + 2, y^4 + 3])
            sage: g = k.absolute_generator(); g
            a0 - a1
            sage: g.minpoly()
            x^2 + 2*a1*x + a1^2 + 2
            sage: g.absolute_minpoly()
            x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
        """
        try:
            return self.__abs_gen
        except AttributeError:
            self.__abs_gen = self._element_class(self, QQ['x'].gen())
            return self.__abs_gen

    def absolute_field(self, names):
        """
        Return ``self`` as an absolute number field.

        INPUT:

        - ``names`` -- string; name of generator of the absolute field

        OUTPUT:

        An absolute number field `K` that is isomorphic to this field.

        Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where
        ``from_K`` is an isomorphism from `K` to ``self`` and ``to_K``
        is an isomorphism from ``self`` to `K`.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: L.<xyz> = K.absolute_field(); L
            Number Field in xyz with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
            sage: L.<c> = K.absolute_field(); L
            Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49

            sage: from_L, to_L = L.structure()
            sage: from_L
            Isomorphism map:
              From: Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
              To:   Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: from_L(c)
            a - b
            sage: to_L
            Isomorphism map:
              From: Number Field in a with defining polynomial x^4 + 3 over its base field
              To:   Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
            sage: to_L(a)
            -5/182*c^7 - 87/364*c^5 - 185/182*c^3 + 323/364*c
            sage: to_L(b)
            -5/182*c^7 - 87/364*c^5 - 185/182*c^3 - 41/364*c
            sage: to_L(a)^4
            -3
            sage: to_L(b)^2
            -2
        """
        return NumberField(self.absolute_polynomial(), names, structure=structure.AbsoluteFromRelative(self))

    def absolute_polynomial_ntl(self):
        """
        Return defining polynomial of this number field
        as a pair, an ntl polynomial and a denominator.

        This is used mainly to implement some internal arithmetic.

        EXAMPLES::

            sage: NumberField(x^2 + (2/3)*x - 9/17,'a').absolute_polynomial_ntl()
            ([-27 34 51], 51)
        """
        try:
            return (self.__abs_polynomial_ntl, self.__abs_denominator_ntl)
        except AttributeError:
            self.__abs_denominator_ntl = ntl.ZZ()
            den = self.absolute_polynomial().denominator()
            self.__abs_denominator_ntl.set_from_sage_int(ZZ(den))
            self.__abs_polynomial_ntl = ntl.ZZX((self.absolute_polynomial()*den).list())
        return (self.__abs_polynomial_ntl, self.__abs_denominator_ntl)

    @cached_method
    def absolute_polynomial(self):
        r"""
        Return the polynomial over `\QQ` that defines this field as an
        extension of the rational numbers.

        .. NOTE::

            The absolute polynomial of a relative number field is
            chosen to be equal to the defining polynomial of the
            underlying PARI absolute number field (it cannot be
            specified by the user).  In particular, it is always a
            monic polynomial with integral coefficients.  On the other
            hand, the defining polynomial of an absolute number field
            and the relative polynomial of a relative number field are
            in general different from their PARI counterparts.

        EXAMPLES::

            sage: k.<a, b> = NumberField([x^2 + 1, x^3 + x + 1]); k
            Number Field in a with defining polynomial x^2 + 1 over its base field
            sage: k.absolute_polynomial()
            x^6 + 5*x^4 - 2*x^3 + 4*x^2 + 4*x + 1

        An example comparing the various defining polynomials to their
        PARI counterparts::

            sage: k.<a, c> = NumberField([x^2 + 1/3, x^2 + 1/4])
            sage: k.absolute_polynomial()
            x^4 - x^2 + 1
            sage: k.pari_polynomial()
            x^4 - x^2 + 1
            sage: k.base_field().absolute_polynomial()
            x^2 + 1/4
            sage: k.pari_absolute_base_polynomial()
            y^2 + 1
            sage: k.relative_polynomial()
            x^2 + 1/3
            sage: k.pari_relative_polynomial()
            x^2 + Mod(y, y^2 + 1)*x - 1
        """
        return QQ['x'](self._pari_rnfeq()[0])

    def relative_polynomial(self):
        """
        Return the defining polynomial of this relative number field over its base field.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.relative_polynomial()
            x^2 + x + 1

        Use absolute polynomial for a polynomial that defines the absolute
        extension.::

            sage: K.absolute_polynomial()
            x^6 + 3*x^5 + 8*x^4 + 9*x^3 + 7*x^2 + 6*x + 3
        """
        return self.__relative_polynomial

    def defining_polynomial(self):
        """
        Return the defining polynomial of this relative number field.

        This is exactly the same as ``relative_polynomial()``.

        EXAMPLES::

            sage: C.<z> = CyclotomicField(5)
            sage: PC.<X> = C[]
            sage: K.<a> = C.extension(X^2 + X + z); K
            Number Field in a with defining polynomial X^2 + X + z over its base field
            sage: K.defining_polynomial()
            X^2 + X + z
        """
        return self.relative_polynomial()

    def polynomial(self):
        """
        For a relative number field, ``polynomial()`` is deliberately
        not implemented.  Either :meth:`~relative_polynomial` or
        :meth:`~absolute_polynomial` must be used.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.polynomial()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field L you must use either L.relative_polynomial() or L.absolute_polynomial() as appropriate
        """
        raise NotImplementedError("For a relative number field L you must use either L.relative_polynomial() or L.absolute_polynomial() as appropriate")

    def base_field(self):
        """
        Return the base field of this relative number field.

        EXAMPLES::

            sage: k.<a> = NumberField([x^3 + x + 1])
            sage: R.<z> = k[]
            sage: L.<b> = NumberField(z^3 + a)
            sage: L.base_field()
            Number Field in a with defining polynomial x^3 + x + 1
            sage: L.base_field() is k
            True

        This is very useful because the print representation of
        a relative field doesn't describe the base field.::

            sage: L
            Number Field in b with defining polynomial z^3 + a over its base field
        """
        return self.__base_field

    def base_ring(self):
        """
        This is exactly the same as base_field.

        EXAMPLES::

            sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1])
            sage: k.base_ring()
            Number Field in a1 with defining polynomial x^3 + x + 1
            sage: k.base_field()
            Number Field in a1 with defining polynomial x^3 + x + 1
        """
        return self.base_field()

    def embeddings(self, K):
        r"""
        Compute all field embeddings of the relative number field self
        into the field `K` (which need not even be a number field,
        e.g., it could be the complex numbers). This will return an
        identical result when given `K` as input again.

        If possible, the most natural embedding of self into `K`
        is put first in the list.

        INPUT:

        - ``K`` -- a field

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^3 - 2, x^2+1])
            sage: f = K.embeddings(ComplexField(58)); f
            [
            Relative number field morphism:
              From: Number Field in a with defining polynomial x^3 - 2 over its base field
              To:   Complex Field with 58 bits of precision
              Defn: a |--> -0.62996052494743676 - 1.0911236359717214*I
                    b |--> -1.9428902930940239e-16 + 1.0000000000000000*I,
            ...
            Relative number field morphism:
              From: Number Field in a with defining polynomial x^3 - 2 over its base field
              To:   Complex Field with 58 bits of precision
              Defn: a |--> 1.2599210498948731
                    b |--> -0.99999999999999999*I
            ]
            sage: f[0](a)^3
            2.0000000000000002 - 8.6389229103644993e-16*I
            sage: f[0](b)^2
            -1.0000000000000001 - 3.8857805861880480e-16*I
            sage: f[0](a+b)
            -0.62996052494743693 - 0.091123635971721295*I
        """
        try:
            # this should be concordant with automorphisms
            return self.__embeddings[K]
        except AttributeError:
            self.__embeddings = {}
        except KeyError:
            pass
        L = self.absolute_field('a')
        E = L.embeddings(K)
        v = [self.hom(f, K) for f in E]

        # If there is an embedding that preserves variable names
        # then it is most natural, so we put it first.
        put_natural_embedding_first(v)

        self.__embeddings[K] = Sequence(v, cr=v!=[], immutable=True, check=False, universe=self.Hom(K))
        return self.__embeddings[K]

    def automorphisms(self):
        r"""
        Compute all Galois automorphisms of self over the base field.  This is
        different than computing the embeddings of self into self; there,
        automorphisms that do not fix the base field are considered.

        EXAMPLES::

            sage: K.<a, b> = NumberField([x^2 + 10000, x^2 + x + 50]); K
            Number Field in a with defining polynomial x^2 + 10000 over its base field
            sage: K.automorphisms()
            [
            Relative number field endomorphism of Number Field in a with defining polynomial x^2 + 10000 over its base field
              Defn: a |--> a
                    b |--> b,
            Relative number field endomorphism of Number Field in a with defining polynomial x^2 + 10000 over its base field
              Defn: a |--> -a
                    b |--> b
            ]
            sage: rho, tau = K.automorphisms()
            sage: tau(a)
            -a
            sage: tau(b) == b
            True

            sage: L.<b, a> = NumberField([x^2 + x + 50, x^2 + 10000, ]); L
            Number Field in b with defining polynomial x^2 + x + 50 over its base field
            sage: L.automorphisms()
            [
            Relative number field endomorphism of Number Field in b with defining polynomial x^2 + x + 50 over its base field
              Defn: b |--> b
                    a |--> a,
            Relative number field endomorphism of Number Field in b with defining polynomial x^2 + x + 50 over its base field
              Defn: b |--> -b - 1
                    a |--> a
            ]
            sage: rho, tau = L.automorphisms()
            sage: tau(a) == a
            True
            sage: tau(b)
            -b - 1

            sage: PQ.<X> = QQ[]
            sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
            sage: PF.<Y> = F[]
            sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
            sage: K.automorphisms()
            [
            Relative number field endomorphism of Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c |--> c
                    a |--> a
                    b |--> b,
            Relative number field endomorphism of Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field
              Defn: c |--> -c
                    a |--> a
                    b |--> b
            ]
        """
        try:
            return self.__automorphisms
        except AttributeError:
            pass

        L = self.absolute_field('a')
        L_into_self, self_into_L = L.structure()
        aas = L.automorphisms() # absolute automorphisms

        a = self_into_L(self.gen())
        abs_base_gens = [self_into_L(_) for _ in self.base_field().gens()]
        v = sorted([ self.hom([ L_into_self(aa(a)) ]) for aa in aas if all(aa(g) == g for g in abs_base_gens) ])
        put_natural_embedding_first(v)
        self.__automorphisms = Sequence(v, cr = (v != []), immutable=True,
                                        check=False, universe=self.Hom(self))
        return self.__automorphisms

    def places(self, all_complex=False, prec=None):
        """
        Return the collection of all infinite places of self.

        By default, this returns the set of real places as
        homomorphisms into RIF first, followed by a choice of one of
        each pair of complex conjugate homomorphisms into CIF.

        On the other hand, if prec is not None, we simply return places
        into RealField(prec) and ComplexField(prec) (or RDF, CDF if
        prec=53).

        There is an optional flag all_complex, which defaults to False. If
        all_complex is True, then the real embeddings are returned as
        embeddings into CIF instead of RIF.

        EXAMPLES::

            sage: L.<b, c> = NumberFieldTower([x^2 - 5, x^3 + x + 3])
            sage: L.places()
            [Relative number field morphism:
            From: Number Field in b with defining polynomial x^2 - 5 over its base field
            To:   Real Field with 106 bits of precision
            Defn: b |--> -2.236067977499789696409173668937
            c |--> -1.213411662762229634132131377426,
            Relative number field morphism:
            From: Number Field in b with defining polynomial x^2 - 5 over its base field
            To:   Real Field with 106 bits of precision
            Defn: b |--> 2.236067977499789696411548005367
            c |--> -1.213411662762229634130492421800,
            Relative number field morphism:
            From: Number Field in b with defining polynomial x^2 - 5 over its base field
            To:   Complex Field with 53 bits of precision
            Defn: b |--> -2.23606797749979 ...e-1...*I
            c |--> 0.606705831381... - 1.45061224918844*I,
            Relative number field morphism:
            From: Number Field in b with defining polynomial x^2 - 5 over its base field
            To:   Complex Field with 53 bits of precision
            Defn: b |--> 2.23606797749979 - 4.44089209850063e-16*I
            c |--> 0.606705831381115 - 1.45061224918844*I]
        """
        L = self.absolute_field('a')
        pl = L.places(all_complex, prec)
        return [self.hom(p, p.codomain()) for p in pl]

    def absolute_different(self):
        r"""
        Return the absolute different of this relative number field `L`, as an
        ideal of `L`. To get the relative different of `L/K`, use
        ``L.relative_different()``.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2 + 1)
            sage: t = K['t'].gen()
            sage: L.<b> = K.extension(t^4 - i)
            sage: L.absolute_different()
            Fractional ideal (8)
        """
        abs = self.absolute_field('a')
        from_abs = abs.structure()[0]
        return self.ideal([from_abs(g) for g in abs.different().gens()])

    def relative_different(self):
        r"""
        Return the relative different of this extension `L/K` as
        an ideal of `L`.  If you want the absolute different of
        `L/\QQ`, use ``L.absolute_different()``.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2 + 1)
            sage: PK.<t> = K[]
            sage: L.<a> = K.extension(t^4  - i)
            sage: L.relative_different()
            Fractional ideal (4)
        """
        I = self.absolute_different()
        J = self.ideal(self.base_field().absolute_different().gens())
        return  I/J

    def different(self):
        """
        The different, unqualified, of a relative number field is deliberately
        not implemented, so that a user cannot mistake the absolute different
        for the relative different, or vice versa.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.different()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field you must use relative_different or absolute_different as appropriate
        """
        raise NotImplementedError("For a relative number field you must use relative_different or absolute_different as appropriate")

    def absolute_discriminant(self, v=None):
        r"""
        Return the absolute discriminant of this relative number field
        or if ``v`` is specified, the determinant of the trace pairing
        on the elements of the list ``v``.

        INPUT:

        - ``v`` (optional) -- list of element of this relative number field.

        OUTPUT: Integer if ``v`` is omitted, and Rational otherwise.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2 + 1)
            sage: t = K['t'].gen()
            sage: L.<b> = K.extension(t^4 - i)
            sage: L.absolute_discriminant()
            16777216
            sage: L.absolute_discriminant([(b + i)^j for j in range(8)])
            61911970349056
        """
        abs = self.absolute_field('a')
        if v is not None:
            to_abs = abs.structure()[1]
            v = [to_abs(x) for x in v]
        return abs.discriminant(v=v)

    def relative_discriminant(self):
        r"""
        Return the relative discriminant of this extension `L/K` as an ideal of
        `K`. If you want the (rational) discriminant of `L/\QQ`, use e.g.
        ``L.absolute_discriminant()``.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2 + 1)
            sage: t = K['t'].gen()
            sage: L.<b> = K.extension(t^4 - i)
            sage: L.relative_discriminant()
            Fractional ideal (256)
            sage: PQ.<X> = QQ[]
            sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
            sage: PF.<Y> = F[]
            sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
            sage: K.relative_discriminant() == F.ideal(4*b)
            True

        TESTS:

        Number fields defined by non-monic and non-integral
        polynomials are supported (:trac:`252`)::

            sage: K.<a> = NumberField(x^2 + 1/2)
            sage: L.<b> = K.extension(x^2 - 1/2)
            sage: L.relative_discriminant()
            Fractional ideal (2)
        """
        base = self.base_field()
        nf = base.pari_nf()
        D, d = nf.rnfdisc(self.pari_relative_polynomial())
        return base.ideal(nf.idealhnf(D))

    def discriminant(self):
        """
        The discriminant, unqualified, of a relative number field is deliberately
        not implemented, so that a user cannot mistake the absolute discriminant
        for the relative discriminant, or vice versa.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.discriminant()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate
        """
        raise NotImplementedError("For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate")

    def disc(self):
        """
        The discriminant, unqualified, of a relative number field is deliberately
        not implemented, so that a user cannot mistake the absolute discriminant
        for the relative discriminant, or vice versa.

        EXAMPLES::

            sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
            sage: K.disc()
            Traceback (most recent call last):
            ...
            NotImplementedError: For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate
        """
        raise NotImplementedError("For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate")

    def order(self, *gens, **kwds):
        r"""
        Return the order with given ring generators in the maximal
        order of this number field.

        INPUT:

        - ``gens`` -- list of elements of self; if no generators are given, just
          returns the cardinality of this number field (oo) for consistency.
        - ``check_is_integral`` -- bool (default: True), whether to check that each
          generator is integral.
        - ``check_rank`` -- bool (default: True), whether to check that the ring
          generated by gens is of full rank.
        - ``allow_subfield`` -- bool (default: False), if True and the generators
          do not generate an order, i.e., they generate a subring of smaller
          rank, instead of raising an error, return an order in a smaller
          number field.

        The check_is_integral and check_rank inputs must be given as
        explicit keyword arguments.

        EXAMPLES::

            sage: P.<a,b,c> = QQ[2^(1/2), 2^(1/3), 3^(1/2)]
            sage: R = P.order([a,b,c]); R
            Relative Order in Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field

        The base ring of an order in a relative extension is still `\ZZ`.::

            sage: R.base_ring()
            Integer Ring

        One must give enough generators to generate a ring of finite index
        in the maximal order::

            sage: P.order([a,b])
            Traceback (most recent call last):
            ...
            ValueError: the rank of the span of gens is wrong
        """
        if len(gens) == 0:
            return NumberField_generic.order(self)
        if len(gens) == 1 and isinstance(gens[0], (list, tuple)):
            gens = gens[0]
        gens = [self(x) for x in gens]
        return relative_order_from_ring_generators(gens, **kwds)

    def is_free(self, proof=None):
        r"""
        Determine whether or not `L/K` is free (i.e. if `\mathcal{O}_L` is
        a free `\mathcal{O}_K`-module).

        INPUT:

        - ``proof`` -- default: True

        EXAMPLES::

            sage: x = polygen(QQ)
            sage: K.<a> = NumberField(x^2+6)
            sage: x = polygen(K)
            sage: L.<b> = K.extension(x^2 + 3)    ## extend by x^2+3
            sage: L.is_free()
            False
        """
        proof = proof_flag(proof)
        base_bnf = self._pari_base_bnf(proof)
        if base_bnf.rnfisfree(self.pari_relative_polynomial()) == 1:
            return True
        return False

    def _factor_univariate_polynomial(self, poly, **kwargs):
        """
        Factorisation of univariate polynomials over relative number fields.

        This is called by the ``factor`` method of univariate polynomials.

        EXAMPLES::

            sage: K.<i> = NumberField(x**2+1)
            sage: L.<sqrt2> = K.extension(x*x-2)
            sage: x = polygen(L,'x')
            sage: factor(x**2+8)  # indirect doctest
            (x + 2*i*sqrt2) * (x - 2*i*sqrt2)
        """
        M = self.absolute_field('a')
        from_M, to_M = M.structure()
        g = M['x']([to_M(x) for x in poly.list()])
        F = g.factor()
        S = poly.parent()
        v = [(S([from_M(x) for x in f.list()]), e) for f, e in F]
        return Factorization(v, from_M(F.unit()))

    def lift_to_base(self, element):
        """
        Lift an element of this extension into the base field if possible,
        or raise a ValueError if it is not possible.

        EXAMPLES::

            sage: x = polygen(ZZ)
            sage: K.<a> = NumberField(x^3 - 2)
            sage: R.<y> = K[]
            sage: L.<b> = K.extension(y^2 - a)
            sage: L.lift_to_base(b^4)
            a^2
            sage: L.lift_to_base(b^6)
            2
            sage: L.lift_to_base(355/113)
            355/113
            sage: L.lift_to_base(b)
            Traceback (most recent call last):
            ...
            ValueError: The element b is not in the base field

        TESTS:

        Number fields defined by non-monic and non-integral
        polynomials are supported (:trac:`252`)::

            sage: R.<x> = QQ[]
            sage: K.<a> = NumberField(x^2 + 1/2)
            sage: L.<b> = K.extension(x^2 - a/2)
            sage: L.lift_to_base(b^2)
            1/2*a
        """
        # Convert the element to a PARI polynomial with t_POLMOD
        # coefficients representing elements of the base field.
        r = self._pari_rnfeq()._eltabstorel_lift(self(element)._pari_polynomial('x'))
        # Lift the coefficients and call simplify() to make PARI check
        # which variables really appear in the resulting polynomial
        # (otherwise we always have a polynomial in two variables even
        # though not all variables actually occur).
        r = r.lift().simplify()

        # Special case: check whether the result is simply an integer or rational
        if r.type() in ["t_INT", "t_FRAC"]:
            return self.base_field()(r)
        # Now we should have a polynomial in the variable y.
        # Otherwise we're not in the base field.
        if r.type() != "t_POL" or str(r.variable()) != 'y':
            raise ValueError("The element %s is not in the base field"%element)
        return self.base_field()(r, check=False)

    def relativize(self, alpha, names):
        r"""
        Given an element in self or an embedding of a subfield into self,
        return a relative number field `K` isomorphic to self that is relative
        over the absolute field `\QQ(\alpha)` or the domain of `\alpha`, along
        with isomorphisms from `K` to self and from self to `K`.

        INPUT:

        - ``alpha`` -- an element of self, or an embedding of a subfield into self
        - ``names`` -- name of generator for output field `K`.

        OUTPUT: `K` -- a relative number field

        Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where
        ``from_K`` is an isomorphism from `K` to self and ``to_K`` is
        an isomorphism from self to `K`.

        EXAMPLES::

            sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
            Number Field in a with defining polynomial x^4 + 3 over its base field
            sage: L.<z,w> = K.relativize(a^2)
            sage: z^2
            z^2
            sage: w^2
            -3
            sage: L
            Number Field in z with defining polynomial x^4 + (-2*w + 4)*x^2 + 4*w + 1 over its base field
            sage: L.base_field()
            Number Field in w with defining polynomial x^2 + 3

        Now suppose we have `K` below `L` below `M`::

            sage: M = NumberField(x^8 + 2, 'a'); M
            Number Field in a with defining polynomial x^8 + 2
            sage: L, L_into_M, _ = M.subfields(4)[0]; L
            Number Field in a0 with defining polynomial x^4 + 2
            sage: K, K_into_L, _ = L.subfields(2)[0]; K
            Number Field in a0_0 with defining polynomial x^2 + 2
            sage: K_into_M = L_into_M * K_into_L

            sage: L_over_K = L.relativize(K_into_L, 'c'); L_over_K
            Number Field in c with defining polynomial x^2 + a0_0 over its base field
            sage: L_over_K_to_L, L_to_L_over_K = L_over_K.structure()
            sage: M_over_L_over_K = M.relativize(L_into_M * L_over_K_to_L, 'd'); M_over_L_over_K
            Number Field in d with defining polynomial x^2 + c over its base field
            sage: M_over_L_over_K.base_field() is L_over_K
            True

        Test relativizing a degree 6 field over its degree 2 and degree 3
        subfields, using both an explicit element::

            sage: K.<a> = NumberField(x^6 + 2); K
            Number Field in a with defining polynomial x^6 + 2
            sage: K2, K2_into_K, _ = K.subfields(2)[0]; K2
            Number Field in a0 with defining polynomial x^2 + 2
            sage: K3, K3_into_K, _ = K.subfields(3)[0]; K3
            Number Field in a0 with defining polynomial x^3 - 2

        Here we explicitly relativize over an element of K2 (not the
        generator)::

            sage: L = K.relativize(K3_into_K, 'b'); L
            Number Field in b with defining polynomial x^2 + a0 over its base field
            sage: L_to_K, K_to_L = L.structure()
            sage: L_over_K2 = L.relativize(K_to_L(K2_into_K(K2.gen() + 1)), 'c'); L_over_K2
            Number Field in c0 with defining polynomial x^3 - c1 + 1 over its base field
            sage: L_over_K2.base_field()
            Number Field in c1 with defining polynomial x^2 - 2*x + 3

        Here we use a morphism to preserve the base field information::

            sage: K2_into_L = K_to_L * K2_into_K
            sage: L_over_K2 = L.relativize(K2_into_L, 'c'); L_over_K2
            Number Field in c with defining polynomial x^3 - a0 over its base field
            sage: L_over_K2.base_field() is K2
            True
        """
        K = self.absolute_field('a')
        from_K, to_K = K.structure()

        if is_Map(alpha):
            # alpha is an embedding of a subfield into self; compose to get an
            # embedding of a subfield into the absolute field
            beta = to_K * alpha
        else:
            # alpha is an element coercible into self
            beta = to_K(alpha)

        L = K.relativize(beta, names)
        return K.relativize(beta, names, structure=structure.RelativeFromRelative(L))

    def uniformizer(self, P, others = "positive"):
        """
        Returns an element of self with valuation 1 at the prime ideal P.

        INPUT:


        -  ``self`` - a number field

        -  ``P`` - a prime ideal of self

        -  ``others`` - either "positive" (default), in which
           case the element will have non-negative valuation at all other
           primes of self, or "negative", in which case the element will have
           non-positive valuation at all other primes of self.


        .. note::

           When P is principal (e.g. always when self has class number
           one) the result may or may not be a generator of P!

        EXAMPLES::

            sage: K.<a, b> = NumberField([x^2 + 23, x^2 - 3])
            sage: P = K.prime_factors(5)[0]; P
            Fractional ideal (5, 1/2*a + b - 5/2)
            sage: u = K.uniformizer(P)
            sage: u.valuation(P)
            1
            sage: (P, 1) in K.factor(u)
            True
        """
        if not is_NumberFieldIdeal(P):
            P = self.ideal(P)
        if not P.is_maximal():
            raise ValueError("P (=%s) must be a nonzero prime."%P)
        abs = self.absolute_field('a')
        from_abs = abs.structure()[0]
        return from_abs(abs.uniformizer(P.absolute_ideal(), others=others))


def NumberField_relative_v1(base_field, poly, name, latex_name, canonical_embedding=None):
    """
    Used for unpickling old pickles.

    EXAMPLES::

        sage: from sage.rings.number_field.number_field_rel import NumberField_relative_v1
        sage: R.<x> = CyclotomicField(3)[]
        sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a')
        Number Field in a with defining polynomial x^2 + 7 over its base field
    """
    return NumberField(poly.change_ring(base_field), name, check=False,
                       embedding=canonical_embedding, latex_name=latex_name)

NumberField_extension_v1 = NumberField_relative_v1  # historical reasons only
Relative Number Fields  AUTHORS:  - <NAME> (2004, 2005): initial version - <NAME> (2006-05-12): added support for relative extensions - <NAME> (2007-09-04): major rewrite and documentation - <NAME> (2008-10): specified embeddings into ambient fields - <NAME> (2009-01): modernize coercion implementation - <NAME> (2012-08): added is_CM_extension - <NAME> (2014-04): absolute number fields are unique parents  This example follows one in the Magma reference manual::      sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000)     sage: z = y^5/11; z     420/11*y^3 - 40000/11*y     sage: R.<y> = PolynomialRing(K)     sage: f = y^2 + y + 1     sage: L.<a> = K.extension(f); L     Number Field in a with defining polynomial y^2 + y + 1 over its base field     sage: KL.<b> = NumberField([x^4 - 420*x^2 + 40000, x^2 + x + 1]); KL     Number Field in b0 with defining polynomial x^4 - 420*x^2 + 40000 over its base field  We do some arithmetic in a tower of relative number fields::      sage: K.<cuberoot2> = NumberField(x^3 - 2)     sage: L.<cuberoot3> = K.extension(x^3 - 3)     sage: S.<sqrt2> = L.extension(x^2 - 2)     sage: S     Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field     sage: sqrt2 * cuberoot3     cuberoot3*sqrt2     sage: (sqrt2 + cuberoot3)^5     (20*cuberoot3^2 + 15*cuberoot3 + 4)*sqrt2 + 3*cuberoot3^2 + 20*cuberoot3 + 60     sage: cuberoot2 + cuberoot3     cuberoot3 + cuberoot2     sage: cuberoot2 + cuberoot3 + sqrt2     sqrt2 + cuberoot3 + cuberoot2     sage: (cuberoot2 + cuberoot3 + sqrt2)^2     (2*cuberoot3 + 2*cuberoot2)*sqrt2 + cuberoot3^2 + 2*cuberoot2*cuberoot3 + cuberoot2^2 + 2     sage: cuberoot2 + sqrt2     sqrt2 + cuberoot2     sage: a = S(cuberoot2); a     cuberoot2     sage: a.parent()     Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field  WARNING: Doing arithmetic in towers of relative fields that depends on canonical coercions is currently VERY SLOW.  It is much better to explicitly coerce all elements into a common field, then do arithmetic with them there (which is quite fast).  TESTS::      sage: y = polygen(QQ,'y'); K.<beta> = NumberField([y^3 - 3, y^2 - 2])     sage: K(y^10)     27*beta0     sage: beta^10     27*beta0 # **************************************************************************** #       Copyright (C) 2004-2009 <NAME> <<EMAIL>> #                     2014-2022 <NAME> <<EMAIL>> # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 2 of the License, or # (at your option) any later version. #                  https://www.gnu.org/licenses/ # **************************************************************************** Return ``True`` if `x` is a relative number field.      EXAMPLES::          sage: from sage.rings.number_field.number_field_rel import is_RelativeNumberField         sage: is_RelativeNumberField(NumberField(x^2+1,'a'))         False         sage: k.<a> = NumberField(x^3 - 2)         sage: l.<b> = k.extension(x^3 - 3); l         Number Field in b with defining polynomial x^3 - 3 over its base field         sage: is_RelativeNumberField(l)         True         sage: is_RelativeNumberField(QQ)         False INPUT:      - ``base`` -- the base field      - ``polynomial`` -- a polynomial which must be defined in the ring `K[x]`,       where `K` is the base field.      - ``name`` -- a string, the variable name      - ``latex_name`` -- a string or ``None`` (default: ``None``), variable name       for latex printing      - ``check`` -- a boolean (default: ``True``), whether to check       irreducibility of ``polynomial``      - ``embedding`` -- currently not supported, must be ``None``      - ``structure`` -- an instance of :class:`structure.NumberFieldStructure`       or ``None`` (default: ``None``), provides additional information about       this number field, e.g., the absolute number field from which it was       created      EXAMPLES::          sage: K.<a> = NumberField(x^3 - 2)         sage: t = polygen(K)         sage: L.<b> = K.extension(t^2+t+a); L         Number Field in b with defining polynomial x^2 + x + a over its base field      TESTS::          sage: Z = var('Z')         sage: K.<w> = NumberField(Z^3 + Z + 1)         sage: L.<z> = K.extension(Z^3 + 2)         sage: K = loads(dumps(L))         sage: K         Number Field in z with defining polynomial Z^3 + 2 over its base field         sage: L is K         True          sage: M.<u,v> = L.change_names()         sage: M.structure()         (Isomorphism given by variable name change map:           From: Number Field in u with defining polynomial x^3 + 2 over its base field           To:   Number Field in z with defining polynomial Z^3 + 2 over its base field,          Isomorphism given by variable name change map:           From: Number Field in z with defining polynomial Z^3 + 2 over its base field           To:   Number Field in u with defining polynomial x^3 + 2 over its base field)         sage: loads(dumps(M)) is M         True Initialization.          EXAMPLES::              sage: K.<x> = CyclotomicField(5)[]             sage: W.<a> = NumberField(x^2 + 1)             sage: W             Number Field in a with defining polynomial x^2 + 1 over its base field             sage: type(W)             <class 'sage.rings.number_field.number_field_rel.NumberField_relative_with_category'>          Test that check=False really skips the test::              sage: W.<a> = NumberField(K.cyclotomic_polynomial(5), check=False)             sage: W             Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1 over its base field          A relative extension of a relative extension::              sage: x = polygen(ZZ)             sage: k.<a0,a1> = NumberField([x^2 + 2, x^2 + 1])             sage: l.<b> = k.extension(x^2 + 3)             sage: l             Number Field in b with defining polynomial x^2 + 3 over its base field             sage: l.base_field()             Number Field in a0 with defining polynomial x^2 + 2 over its base field             sage: l.base_field().base_field()             Number Field in a1 with defining polynomial x^2 + 1          Non-monic and non-integral polynomials are supported (:trac:`252`)::              sage: l.<b> = k.extension(5*x^2 + 3); l             Number Field in b with defining polynomial 5*x^2 + 3 over its base field             sage: l.pari_rnf()             [x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4), ..., y^4 + 6*y^2 + 1, x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4)], [0, 0]]             sage: b             b              sage: l.<b> = k.extension(x^2 + 3/5); l             Number Field in b with defining polynomial x^2 + 3/5 over its base field             sage: l.pari_rnf()             [x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4), ..., y^4 + 6*y^2 + 1, x^2 + (-1/2*y^2 + y - 3/2)*x + (-1/4*y^3 + 1/4*y^2 - 3/4*y - 13/4)], [0, 0]]             sage: b             b              sage: l.<b> = k.extension(x - 1/a0); l             Number Field in b with defining polynomial x + 1/2*a0 over its base field             sage: l.pari_rnf()             [x, [4, -x^3 - x^2 - 7*x - 3, -x^3 + x^2 - 7*x + 3, 2*x^3 + 10*x], ..., [x^4 + 6*x^2 + 1, -x, -1, y^4 + 6*y^2 + 1, x], [0, 0]]             sage: b             -1/2*a0          TESTS:          Test that irreducibility testing is working::              sage: x = polygen(ZZ)             sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 3])             sage: K.<a> = NumberField(x^2 + 2)             sage: x = polygen(K)             sage: L.<b> = K.extension(x^3 + 3*a)              sage: (x^3 + 2*a).factor()             (x - a) * (x^2 + a*x - 2)             sage: L.<b> = K.extension(x^3 + 2*a)             Traceback (most recent call last):             ...             ValueError: defining polynomial (x^3 + 2*a) must be irreducible             sage: (x^2 + 2).factor()             (x - a) * (x + a)             sage: L.<b> = K.extension(x^2 + 2)             Traceback (most recent call last):             ...             ValueError: defining polynomial (x^2 + 2) must be irreducible             sage: L.<b> = K.extension(x^2 + 2)             Traceback (most recent call last):             ...             ValueError: defining polynomial (x^2 + 2) must be irreducible          Error checks::              sage: x = polygen(ZZ)             sage: K.<a> = NumberField(x^2 + 1)             sage: K.extension(x^2 + 2, 'a')             Traceback (most recent call last):             ...             ValueError: base field and extension cannot have the same name 'a' #raise ValueError, "The polynomial must be defined over the base field" # Generate the nf and bnf corresponding to the base field # defined as polynomials in y, e.g. for rnfisfree # Convert the polynomial defining the base field into a # polynomial in y to satisfy PARI's ordering requirements. Return relative number field isomorphic to self but with the         given generator names.          INPUT:          - ``names`` -- number of names should be at most the number of           generators of self, i.e., the number of steps in the tower           of relative fields.          Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where         from_K is an isomorphism from `K` to self and ``to_K`` is an         isomorphism from self to `K`.          EXAMPLES::              sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K             Number Field in a with defining polynomial x^4 + 3 over its base field             sage: L.<c,d> = K.change_names()             sage: L             Number Field in c with defining polynomial x^4 + 3 over its base field             sage: L.base_field()             Number Field in d with defining polynomial x^2 + 2          An example with a 3-level tower::              sage: K.<a,b,c> = NumberField([x^2 + 17, x^2 + x + 1, x^3 - 2]); K             Number Field in a with defining polynomial x^2 + 17 over its base field             sage: L.<m,n,r> = K.change_names()             sage: L             Number Field in m with defining polynomial x^2 + 17 over its base field             sage: L.base_field()             Number Field in n with defining polynomial x^2 + x + 1 over its base field             sage: L.base_field().base_field()             Number Field in r with defining polynomial x^3 - 2          And a more complicated example::              sage: PQ.<X> = QQ[]             sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])             sage: PF.<Y> = F[]             sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)             sage: L.<m, n, r> = K.change_names(); L             Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field             sage: L.structure()             (Isomorphism given by variable name change map:               From: Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field               To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field,              Isomorphism given by variable name change map:               From: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field               To:   Number Field in m with defining polynomial x^2 + (-2*r - 3)*n - 2*r - 6 over its base field) Return all subfields of this relative number field self of the given degree,         or of all possible degrees if degree is 0.  The subfields are returned as         absolute fields together with an embedding into self.  For the case of the         field itself, the reverse isomorphism is also provided.          EXAMPLES::              sage: PQ.<X> = QQ[]             sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])             sage: PF.<Y> = F[]             sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)             sage: K.subfields(2)             [             (Number Field in c0 with defining polynomial x^2 - 24*x + 96, Ring morphism:               From: Number Field in c0 with defining polynomial x^2 - 24*x + 96               To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field               Defn: c0 |--> -4*b + 12, None),             (Number Field in c1 with defining polynomial x^2 - 24*x + 120, Ring morphism:               From: Number Field in c1 with defining polynomial x^2 - 24*x + 120               To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field               Defn: c1 |--> 2*b*a + 12, None),             (Number Field in c2 with defining polynomial x^2 - 24*x + 72, Ring morphism:               From: Number Field in c2 with defining polynomial x^2 - 24*x + 72               To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field               Defn: c2 |--> -6*a + 12, None)             ]             sage: K.subfields(8, 'w')             [             (Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9, Ring morphism:               From: Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9               To:   Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field               Defn: w0 |--> (-1/2*b*a + 1/2*b + 1/2)*c, Relative number field morphism:               From: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field               To:   Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 + 9               Defn: c |--> -1/3*w0^7 + 4*w0^5 - 12*w0^3 + 11*w0                     a |--> 1/3*w0^6 - 10/3*w0^4 + 5*w0^2                     b |--> -2/3*w0^6 + 7*w0^4 - 14*w0^2 + 6)             ]             sage: K.subfields(3)             [] Returns False, since this is not an absolute field.          EXAMPLES::              sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K             Number Field in a with defining polynomial x^4 + 3 over its base field             sage: K.is_absolute()             False             sage: K.is_relative()             True Return the generators of this relative number field.          EXAMPLES::              sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K             Number Field in a with defining polynomial x^4 + 3 over its base field             sage: K.gens()             (a, b)          TESTS:          Trivial extensions work like non-trivial ones (:trac:`2220`)::              sage: NumberField([x^2 - 3, x], 'a').gens()             (a0, 0)             sage: NumberField([x, x^2 - 3], 'a').gens()             (0, a1) Return the first `n` generators of this relative number field.          If `n` is greater than the number of generators, the output is         the same as that of :meth:`gens`.          EXAMPLES::              sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K             Number Field in a with defining polynomial x^4 + 3 over its base field             sage: K._first_ngens(0)             ()             sage: K._first_ngens(1)             (a,)             sage: K._first_ngens(2)             (a, b)             sage: K._first_ngens(3)             (a, b) Return the number of generators of this relative number field.          EXAMPLES::              sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K             Number Field in a with defining polynomial x^4 + 3 over its base field             sage: K.gens()             (a, b)             sage: K.ngens()             2 Return the `n`'th generator of this relative number field.          EXAMPLES::              sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K             Number Field in a with defining polynomial x^4 + 3 over its base field             sage: K.gens()             (a, b)             sage: K.gen(0)             a Return the absolute number field `K` that is the Galois closure of this         relative number field.          EXAMPLES::              sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K             Number Field in a with defining polynomial x^4 + 3 over its base field             sage: K.galois_closure('c')             Number Field in c with defining polynomial x^16 + 16*x^14 + 28*x^12 + 784*x^10 + 19846*x^8 - 595280*x^6 + 2744476*x^4 + 3212848*x^2 + 29953729 List of all possible composite number fields formed from self and         other, together with (optionally) embeddings into the compositum;         see the documentation for both_maps below.          Since relative fields do not have ambient embeddings,         preserve_embedding has no effect.  In every case all possible         composite number fields are returned.          INPUT:          - ``other`` - a number field          - ``names`` - generator name for composite fields          - ``both_maps`` - (default: False)  if True, return quadruples           (F, self_into_F, other_into_F, k) such that self_into_F maps self into           F, other_into_F maps other into F.  For relative number fields k is           always None.         - ``preserve_embedding`` - (default: True) has no effect, but is kept           for compatibility with the absolute version of this function.  In every           case the list of all possible compositums is returned.          OUTPUT:          -  ``list`` - list of the composite fields, possibly with maps.           EXAMPLES::              sage: K.<a, b> = NumberField([x^2 + 5, x^2 - 2])             sage: L.<c, d> = NumberField([x^2 + 5, x^2 - 3])             sage: K.composite_fields(L, 'e')             [Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600]             sage: K.composite_fields(L, 'e', both_maps=True)             [[Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600,               Relative number field morphism:               From: Number Field in a with defining polynomial x^2 + 5 over its base field              To:   Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600               Defn: a |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e                     b |--> -21/166400*e^7 + 73/20800*e^5 - 779/10400*e^3 + 7/260*e,               Relative number field morphism:               From: Number Field in c with defining polynomial x^2 + 5 over its base field               To:   Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2 + 25600               Defn: c |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e                     d |--> -3/25600*e^7 + 7/1600*e^5 - 147/1600*e^3 + 1/40*e,               None]] The degree of this relative number field over the rational field.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])             sage: K.absolute_degree()             6 Returns the relative degree of this relative number field.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])             sage: K.relative_degree()             2 The degree, unqualified, of a relative number field is deliberately         not implemented, so that a user cannot mistake the absolute degree         for the relative degree, or vice versa.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])             sage: K.degree()             Traceback (most recent call last):             ...             NotImplementedError: For a relative number field you must use relative_degree or absolute_degree as appropriate Implements :meth:`NumberField_generic.maximal_order` for relative         number fields.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])             sage: K.maximal_order() is K.maximal_order()  # indirect doctest             True Return string representation of this relative number field.          The base field is not part of the string representation.  To         find out what the base field is use :meth:`~base_field`.          EXAMPLES::              sage: k.<a, b> = NumberField([x^5 + 2, x^7 + 3])             sage: repr(k) # indirect doctest             'Number Field in a with defining polynomial x^5 + 2 over its base field'             sage: k.base_field()             Number Field in b with defining polynomial x^7 + 3 Return homset of homomorphisms from this relative number field         to the codomain.          EXAMPLES:          This function is implicitly called by the ``Hom`` method or         function::              sage: K.<a,b> = NumberField([x^3 - 2, x^2+1])             sage: K.Hom(K) # indirect doctest             Automorphism group of Number Field in a with defining polynomial x^3 - 2 over its base field             sage: type(K.Hom(K))             <class 'sage.rings.number_field.homset.RelativeNumberFieldHomset_with_category'>          TESTS::              sage: H = End(K)             sage: loads(dumps(H)) is H             True Return a `\LaTeX` representation of the extension.          EXAMPLES::              sage: x = polygen(QQ)             sage: K.<a> = NumberField(x^3 - 2)             sage: t = polygen(K)             sage: K.extension(t^2+t+a, 'b')._latex_()             '( \\Bold{Q}[a]/(a^{3} - 2) )[b]/(b^{2} + b + a)' Coerce a number field element x into this number field.          In most cases this currently doesn't work (since it is         barely implemented) -- it only works for constants.          INPUT:          - ``x`` -- an element of some number field          EXAMPLES::              sage: K.<a> = NumberField(x^3 + 2)             sage: L.<b> = NumberField(x^2 + 1)             sage: K._coerce_from_other_number_field(L(2/3))             2/3 # todo: more general coercion if embedding have been asserted Convert the non-number field element `x` into this number field.          INPUT:          - ``x`` -- a non number field element, e.g., a list,           integer, rational, or polynomial.          EXAMPLES::              sage: x = polygen(ZZ)             sage: K.<a> = NumberField(x^5 + 2)             sage: L.<b> = K.extension(x^2 + 3*a)             sage: R.<u> = QQ[]             sage: S.<t> = R[]              sage: L(a + b)             b + a              sage: L(5*t*(1 + u) + 2/3*u)             (5*a + 5)*b + 2/3*a             sage: L(0*t + 2/3)             2/3             sage: L(1/2*t + 5)             1/2*b + 5          This seems reasonable::              sage: L(t*5)             5*b          This is misleading, but correct!  It is more often desired         to make a number field element given by rational         coefficients of the relative power basis (so 2*b^2 + 3)         than it is to create the constant term of such an element,         which is what would happen if L(u*5) gave 5*a.::              sage: L(u*5)             5*b              sage: L([1, 1/2])             1/2*b + 1             sage: L([ a, 1/2 + a/3 ])             (1/3*a + 1/2)*b + a              sage: L([ 1 ])             Traceback (most recent call last):             ...             ValueError: Length must be equal to the degree of this number field          TESTS:          Examples from :trac:`4727`::              sage: K.<j,b> = QQ[sqrt(-1), sqrt(2)]             sage: j             I             sage: j.list()             [0, 1]             sage: K(j.list())             I             sage: (b*j + 1/2).list()             [1/2, sqrt2]             sage: K((b*j + 1/2).list())             sqrt2*I + 1/2          Examples from :trac:`4869`::              sage: K.<z> = CyclotomicField(7)             sage: Ky.<y> = PolynomialRing(K)             sage: L.<a> = K.extension(y^2 + 1)             sage: K(K.polynomial_ring().random_element()) # random             -12*z^2 + 1/2*z - 1/95             sage: L(L.polynomial_ring().random_element()) # random             (z^5 + 1/3*z^4 - z^3 + z^2 - z + 2/3)*a + 1/4*z^5 - 7/2*z^4 + 5/3*z^3 - 1/4*z^2 + 3/2*z - 1          Examples from :trac:`11307`::              sage: L = NumberField([x^2 + 1, x^2 - 3], 'a')             sage: L(L)             Traceback (most recent call last):             ...             TypeError: unable to convert Number Field in a0 with defining polynomial x^2 + 1 over its base field to Number Field in a0 with defining polynomial x^2 + 1 over its base field             sage: L in L             False          We construct the composite of three quadratic fields, then         coerce from the quartic subfield of the relative extension::              sage: k.<a,b,c> = NumberField([x^2 + 5, x^2 + 3, x^2 + 1])             sage: m = k.base_field(); m             Number Field in b with defining polynomial x^2 + 3 over its base field             sage: k(m.0)             b             sage: k(2/3)             2/3             sage: k(m.0^4)             9              sage: x = polygen(ZZ)             sage: K.<a> = NumberField(x^2 + 2, 'a')             sage: L.<b> = K.extension(x - a, 'b')             sage: L(a)             a             sage: L(b+a)             2*a             sage: K.<a> = NumberField(x^5 + 2, 'a')             sage: L.<b> = K.extension(x - a, 'b')             sage: L(a)             a             sage: L(a**3)             a^3             sage: L(a**2+b)             a^2 + a             sage: L.<b> = K.extension(x + a/2, 'b')             sage: L(a)             a             sage: L(a).polynomial()             -x             sage: L(a).minpoly()             x - a             sage: L(a).absolute_minpoly()             x^5 + 2             sage: L(b)             -1/2*a             sage: L(b).polynomial()             1/2*x             sage: L(b).absolute_minpoly()             x^5 - 1/16             sage: L(b).minpoly()             x + 1/2*a          ::              sage: K.<a> = NumberField(x^5+2)             sage: R.<y> = K[]             sage: L.<x0> = K.extension(y + a**2)             sage: L(a)             a # we have been given a polynomial, change it to an absolute polynomial # special case absolute polynomials -- they should be # in terms of the relative generator # this should work for base_ring()['x'] and QQ['base']['ext'] # Anything else: use the code for generic number fields Canonical coercion of x into this relative number field.          Currently integers, rationals, the base field, and this field         itself coerce canonically into this field (and hence so does         anything that coerces into one of these).          EXAMPLES::              sage: k.<a> = NumberField([x^5 + 2, x^7 + 3])             sage: b = k(k.base_field().gen())             sage: b = k.coerce(k.base_field().gen()) # indirect doctest             sage: b^7             -3             sage: k.coerce(2/3)             2/3             sage: c = a + b # no output Construct a relative number field element from ``x``.          EXAMPLES:          We can create relative number field elements from PARI::              sage: y = polygen(QQ)             sage: K.<a> = NumberField(y^2 + y + 1)             sage: x = polygen(K)             sage: L.<b> = NumberField(x^4 + a*x + 2)             sage: e = a.__pari__(); e             Mod(y, y^2 + y + 1)             sage: L(e)  # Conversion from PARI base field element             a             sage: e = (a*b).__pari__('x'); e             Mod(-x^4 - 2, x^8 - x^5 + 4*x^4 + x^2 - 2*x + 4)             sage: L(e)  # Conversion from PARI absolute number field element             a*b             sage: e = L.pari_rnf().rnfeltabstorel(e); e             Mod(Mod(y, y^2 + y + 1)*x, x^4 + Mod(y, y^2 + y + 1)*x + 2)             sage: L(e)  # Conversion from PARI relative number field element             a*b             sage: e = pari('Mod(0, x^8 + 1)'); L(e)  # Wrong modulus             Traceback (most recent call last):             ...             TypeError: cannot convert PARI element Mod(0, x^8 + 1) into Number Field in b with defining polynomial x^4 + a*x + 2 over its base field          We test a relative number field element created "by hand"::              sage: e = pari("Mod(Mod(y, y^2 + y + 1)*x^2 + Mod(1, y^2 + y + 1), x^4 + y*x + 2)")             sage: L(e)             a*b^2 + 1          A wrong modulus yields an error::              sage: e = pari('Mod(y*x, x^4 + y^2*x + 2)'); L(e)             Traceback (most recent call last):             ...             TypeError: cannot convert PARI element Mod(y*x, x^4 + y^2*x + 2) into Number Field in b with defining polynomial x^4 + a*x + 2 over its base field # If x is a *relative* PARI number field element, convert it # to an absolute element. Given an element of the base field, give its inclusion into         this extension in terms of the generator of this field.          This is called by the canonical coercion map on elements from         the base field.          EXAMPLES::              sage: k.<a> = NumberField([x^2 + 3, x^2 + 1])             sage: m = k.base_field(); m             Number Field in a1 with defining polynomial x^2 + 1             sage: k.coerce(m.0 + 2/3) # indirect doctest             a1 + 2/3             sage: s = k.coerce(m.0); s             a1             sage: s^2             -1          This implicitly tests this coercion map::              sage: K.<a> = NumberField([x^2 + p for p in [5,3,2]])             sage: K.coerce(K.base_field().0)             a1             sage: K.coerce(K.base_field().0)^2             -3          TESTS:          Check that :trac:`5828` is solved::              sage: K.<w> = QuadraticField(-1)             sage: KX.<X> = K[]             sage: H.<h> = K.extension(X-1)             sage: H(w)             w # Write element in terms of the absolute base field # Express element as a polynomial in the absolute generator of self # We do NOT call self(...) because this code is called by # __init__ before we initialize self.gens(), and self(...) # uses self.gens() Return the Python class used to represent ideals of a relative         number field.          EXAMPLES::              sage: k.<a> = NumberField([x^5 + 2, x^7 + 3])             sage: k._fractional_ideal_class_ ()             <class 'sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel'> Return the PARI bnf (big number field) representation of the         absolute base field in terms of the pari variable ``y``, suitable         for extension by the pari variable ``x``.          All caching is done by the absolute base field.          INPUT:          - ``proof`` (bool, default True) -- if True, certify           correctness of calculations (not assuming GRH).          EXAMPLES::              sage: k.<a> = NumberField([x^3 + 2, x^2 + 2])             sage: k._pari_base_bnf()             [[;], matrix(0,3), [;], ...] Return the PARI number field representation of the absolute         base field, in terms of the pari variable ``y``, suitable for         extension by the pari variable ``x``.          All caching is done by the absolute base field.          EXAMPLES::              sage: y = polygen(QQ,'y')             sage: k.<a> = NumberField([y^3 + 2, y^2 + 2])             sage: k._pari_base_nf()             [y^2 + 2, [0, 1], -8, 1, ..., [1, 0, 0, -2; 0, 1, 1, 0]] For a relative number field, ``is_galois()`` is deliberately not         implemented, since it is not clear whether this would mean "Galois over         `\QQ`" or "Galois over the given base field". Use either ``is_galois_absolute()`` or ``is_galois_relative()`` respectively.          EXAMPLES::              sage: k.<a> =NumberField([x^3 - 2, x^2 + x + 1])             sage: k.is_galois()             Traceback (most recent call last):             ...             NotImplementedError: For a relative number field L you must use either L.is_galois_relative() or L.is_galois_absolute() as appropriate Return True if for this relative extension `L/K`, `L` is a         Galois extension of `K`.          EXAMPLES::              sage: K.<a> = NumberField(x^3 - 2)             sage: y = polygen(K)             sage: L.<b> = K.extension(y^2 - a)             sage: L.is_galois_relative()             True             sage: M.<c> = K.extension(y^3 - a)             sage: M.is_galois_relative()             False          The next example previously gave a wrong result; see :trac:`9390`::              sage: F.<a, b> = NumberField([x^2 - 2, x^2 - 3])             sage: F.is_galois_relative()             True Return True if for this relative extension `L/K`, `L` is a Galois extension of `\QQ`.          EXAMPLES::              sage: K.<a> = NumberField(x^3 - 2)             sage: y = polygen(K); L.<b> = K.extension(y^2 - a)             sage: L.is_galois_absolute()             False For this relative extension `L/K` and another relative extension `M/K`, return True         if there is a `K`-linear isomorphism from `L` to `M`. More generally, ``other`` can be a         relative extension `M/K^\prime` with ``base_isom`` an isomorphism from `K` to         `K^\prime`.          EXAMPLES::              sage: K.<z9> = NumberField(x^6 + x^3 + 1)             sage: R.<z> = PolynomialRing(K)             sage: m1 = 3*z9^4 - 4*z9^3 - 4*z9^2 + 3*z9 - 8             sage: L1 = K.extension(z^2 - m1, 'b1')             sage: G = K.galois_group(); gamma = G.gen()             sage: m2 = (gamma^2)(m1)             sage: L2 = K.extension(z^2 - m2, 'b2')             sage: L1.is_isomorphic_relative(L2)             False             sage: L1.is_isomorphic(L2)             True             sage: L3 = K.extension(z^4 - m1, 'b3')             sage: L1.is_isomorphic_relative(L3)             False          If we have two extensions over different, but isomorphic, bases, we can compare them by         letting ``base_isom`` be an isomorphism from self's base field to other's base field::              sage: Kcyc.<zeta9> = CyclotomicField(9)             sage: Rcyc.<zcyc> = PolynomialRing(Kcyc)             sage: phi1 = K.hom([zeta9])             sage: m1cyc = phi1(m1)             sage: L1cyc = Kcyc.extension(zcyc^2 - m1cyc, 'b1cyc')             sage: L1.is_isomorphic_relative(L1cyc, base_isom=phi1)             True             sage: L2.is_isomorphic_relative(L1cyc, base_isom=phi1)             False             sage: phi2 = K.hom([phi1((gamma^(-2))(z9))])             sage: L1.is_isomorphic_relative(L1cyc, base_isom=phi2)             False             sage: L2.is_isomorphic_relative(L1cyc, base_isom=phi2)             True          Omitting ``base_isom`` raises a ValueError when the base fields are not identical::              sage: L1.is_isomorphic_relative(L1cyc)             Traceback (most recent call last):             ...             ValueError: other does not have the same base field as self, so an isomorphism from self's base_field to other's base_field must be provided using the base_isom parameter.          The parameter ``base_isom`` can also be used to check if the relative extensions are         Galois conjugate::              sage: for g in G:             ....:   if L1.is_isomorphic_relative(L2, g.as_hom()):             ....:       print(g.as_hom())             Ring endomorphism of Number Field in z9 with defining polynomial x^6 + x^3 + 1               Defn: z9 |--> z9^4 Return True is this is a CM extension, i.e. a totally imaginary         quadratic extension of a totally real field.          EXAMPLES::              sage: F.<a> = NumberField(x^2 - 5)             sage: K.<z> = F.extension(x^2 + 7)             sage: K.is_CM_extension()             True             sage: K = CyclotomicField(7)             sage: K_rel = K.relativize(K.gen() + K.gen()^(-1), 'z')             sage: K_rel.is_CM_extension()             True             sage: F = CyclotomicField(3)             sage: K.<z> = F.extension(x^3 - 2)             sage: K.is_CM_extension()             False          A CM field K such that K/F is not a CM extension          ::              sage: F.<a> = NumberField(x^2 + 1)             sage: K.<z> = F.extension(x^2 - 3)             sage: K.is_CM_extension()             False             sage: K.is_CM()             True Return a vector space over a specified subfield that is isomorphic to this number field,         together with the isomorphisms in each direction.          INPUT:          - ``base`` -- a subfield          - ``basis`` -- (optional) a list of elements giving a basis over the subfield          - ``map`` -- (default ``True``) whether to return isomorphisms to and from the vector space          EXAMPLES::              sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K             Number Field in a with defining polynomial x^2 + 2 over its base field             sage: V, from_V, to_V = K.free_module()             sage: to_V(K.0)             (0, 1)             sage: W, from_W, to_W = K.free_module(base=QQ)             sage: w = to_W(K.0); len(w)             18             sage: w[0]             -127917622658689792301282/48787705559800061938765 Return vector space over the base field of self and isomorphisms         from the vector space to self and in the other direction.          EXAMPLES::              sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K             Number Field in a with defining polynomial x^2 + 2 over its base field             sage: V, from_V, to_V = K.relative_vector_space()             sage: from_V(V.0)             1             sage: to_V(K.0)             (0, 1)             sage: from_V(to_V(K.0))             a             sage: to_V(from_V(V.0))             (1, 0)             sage: to_V(from_V(V.1))             (0, 1)          The underlying vector space and maps is cached::              sage: W, from_V, to_V = K.relative_vector_space()             sage: V is W             True Return vector space over `\QQ` of ``self`` and isomorphisms from         the vector space to ``self`` and in the other direction.          EXAMPLES::              sage: K.<a,b> = NumberField([x^3 + 3, x^3 + 2]); K             Number Field in a with defining polynomial x^3 + 3 over its base field             sage: V,from_V,to_V = K.absolute_vector_space(); V             Vector space of dimension 9 over Rational Field             sage: from_V             Isomorphism map:               From: Vector space of dimension 9 over Rational Field               To:   Number Field in a with defining polynomial x^3 + 3 over its base field             sage: to_V             Isomorphism map:               From: Number Field in a with defining polynomial x^3 + 3 over its base field               To:   Vector space of dimension 9 over Rational Field             sage: c = (a+1)^5; c             7*a^2 - 10*a - 29             sage: to_V(c)             (-29, -712/9, 19712/45, 0, -14/9, 364/45, 0, -4/9, 119/45)             sage: from_V(to_V(c))             7*a^2 - 10*a - 29             sage: from_V(3*to_V(b))             3*b For a relative number field, ``vector_space()`` is         deliberately not implemented, so that a user cannot confuse         :meth:`~relative_vector_space` with :meth:`~absolute_vector_space`.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])             sage: K.vector_space()             Traceback (most recent call last):             ...             NotImplementedError: For a relative number field L you must use either L.relative_vector_space() or L.absolute_vector_space() as appropriate Return the base field of this relative extension, but viewed         as an absolute field over `\QQ`.          EXAMPLES::              sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 3, x^3 + 2])             sage: K             Number Field in a with defining polynomial x^2 + 2 over its base field             sage: K.base_field()             Number Field in b with defining polynomial x^3 + 3 over its base field             sage: K.absolute_base_field()[0]             Number Field in a0 with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1             sage: K.base_field().absolute_field('z')             Number Field in z with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1 Return PARI data attached to this relative number field.          OUTPUT:          A 5-element PARI vector containing an absolute polynomial and         further data needed for ``eltabstorel`` and ``eltreltoabs``,         obtained from PARI's ``nf_rnfeq`` function.  This means that         we do not have to initialize a full PARI ``rnf`` structure.          TESTS::              sage: K.<a> = NumberField(x^2 + 2)             sage: x = polygen(K)             sage: L.<b> = K.extension(x^5 + 2*a)             sage: L._pari_rnfeq()             [x^10 + 8, -1/2*x^5, 0, y^2 + 2, x^5 + 2*y]             sage: x = polygen(ZZ)             sage: NumberField(x^10 + 8, 'a').is_isomorphic(L)             True          Initialization is lazy enough to allow arithmetic in massive fields::              sage: K.<a> = NumberField(x^10 + 2000*x + 100001)             sage: x = polygen(K)             sage: L.<b> = K.extension(x^10 + 2*a)             sage: L._pari_rnfeq()             [x^100 - 1024000*x^10 + 102401024, -1/2*x^10, 0, y^10 + 2000*y + 100001, x^10 + 2*y]             sage: a + b             b + a             sage: b^100             -2048000*a - 102401024             sage: (-2*a)^10             -2048000*a - 102401024 Return PARI data needed for constructing relative number field         elements from elements of the base field.          TESTS::              sage: K.<a> = NumberField(x^2 - 2)             sage: L.<b> = K.extension(x^2 - 3)             sage: L._pari_nfzk()             [2, -x^3 + 9*x] Return data relating the Sage and PARI relative polynomials.          OUTPUT:          Let `L` be this relative number field, let `K` be its base         field, and let `f` be the defining polynomial of `L` over `K`.         This method returns a triple ``(g, alpha, beta)``, where          - ``g`` is the defining relative polynomial of the PARI           ``rnf`` structure (see :meth:`pari_rnf`);          - ``alpha`` is the image of `x \bmod f` under some isomorphism           `\phi\colon K[x]/(f) \to K[x]/(g)`;          - ``beta`` is the image of `x \bmod g` under the inverse           isomorphism `\phi^{-1}\colon K[x]/(g) \to K[x]/(f)`.          EXAMPLES::          If the defining polynomials are monic and integral, the result         satisfies ``g = f`` and ``alpha = beta = x``::              sage: R.<x> = QQ[]             sage: K.<a> = NumberField(x^2 + 1)             sage: L.<b> = K.extension(x^2 - a)             sage: L._pari_relative_structure()             (Mod(1, y^2 + 1)*x^2 + Mod(-y, y^2 + 1),              Mod(x, Mod(1, y^2 + 1)*x^2 + Mod(-y, y^2 + 1)),              Mod(x, Mod(1, y^2 + 1)*x^2 + Mod(-y, y^2 + 1)))          An example where the base field is defined by a monic integral         polynomial, but the extension is not::              sage: K.<a> = NumberField(x^2 + 1)             sage: L.<b> = K.extension(x^2 - 1/2)             sage: L._pari_relative_structure()             (x^2 + Mod(-y, y^2 + 1),              Mod(Mod(1/2*y - 1/2, y^2 + 1)*x, x^2 + Mod(-y, y^2 + 1)),              Mod(Mod(-y - 1, y^2 + 1)*x, Mod(1, y^2 + 1)*x^2 + Mod(-1/2, y^2 + 1)))          An example where both fields are defined by non-integral or         non-monic polynomials::              sage: K.<a> = NumberField(2*x^2 + 1)             sage: L.<b> = K.extension(x^2 - 1/3)             sage: L._pari_relative_structure()             (x^2 + Mod(y, y^2 + 2)*x + 1,              Mod(Mod(-1/3*y, y^2 + 2)*x + Mod(1/3, y^2 + 2), x^2 + Mod(y, y^2 + 2)*x + 1),              Mod(Mod(3/2*y, y^2 + 2)*x + Mod(-1/2*y, y^2 + 2), Mod(1, y^2 + 2)*x^2 + Mod(-1/3, y^2 + 2)))          Note that in the last example, the *absolute* defining         polynomials is the same for Sage and PARI, even though this is         not the case for the base field::              sage: K._pari_absolute_structure()             (y^2 + 2, Mod(1/2*y, y^2 + 2), Mod(2*y, y^2 + 1/2))             sage: L._pari_absolute_structure()             (y^4 + 4*y^2 + 1, Mod(y, y^4 + 4*y^2 + 1), Mod(y, y^4 + 4*y^2 + 1)) # PARI's rnfpolredbest() does not always return a # polynomial with integral coefficients in this case. Return root of defining polynomial, which is a generator of         the relative number field over the base.          EXAMPLES::              sage: k.<a> = NumberField(x^2+1); k             Number Field in a with defining polynomial x^2 + 1             sage: y = polygen(k)             sage: m.<b> = k.extension(y^2+3); m             Number Field in b with defining polynomial x^2 + 3 over its base field             sage: c = m.gen(); c # indirect doctest             b             sage: c^2 + 3             0          An example where the defining polynomials are not monic or         integral::              sage: K.<a> = NumberField(3*x^2 + 1)             sage: L.<b> = K.extension(x^2 - 1/2)             sage: L._gen_relative()             b Return the PARI relative number field object associated         to this relative extension.          EXAMPLES::              sage: k.<a> = NumberField([x^4 + 3, x^2 + 2])             sage: k.pari_rnf()             [x^4 + 3, [364, -10*x^7 - 87*x^5 - 370*x^3 - 41*x], [108, 3], ...] Return the PARI polynomial defining the absolute base field, in ``y``.          EXAMPLES::              sage: x = polygen(ZZ)             sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 3]); K             Number Field in a with defining polynomial x^2 + 2 over its base field             sage: K.pari_absolute_base_polynomial()             y^2 + 3             sage: type(K.pari_absolute_base_polynomial())             <class 'cypari2.gen.Gen'>             sage: z = ZZ['z'].0             sage: K.<a, b, c> = NumberField([z^2 + 2, z^2 + 3, z^2 + 5]); K             Number Field in a with defining polynomial z^2 + 2 over its base field             sage: K.pari_absolute_base_polynomial()             y^4 + 16*y^2 + 4             sage: K.base_field()             Number Field in b with defining polynomial z^2 + 3 over its base field             sage: len(QQ['y'](K.pari_absolute_base_polynomial()).roots(K.base_field()))             4             sage: type(K.pari_absolute_base_polynomial())             <class 'cypari2.gen.Gen'> Return the PARI relative polynomial associated to this number         field.          This is always a polynomial in x and y, suitable for PARI's         rnfinit function.  Notice that if this is a relative extension         of a relative extension, the base field is the absolute base         field.          EXAMPLES::              sage: k.<i> = NumberField(x^2 + 1)             sage: m.<z> = k.extension(k['w']([i,0,1]))             sage: m             Number Field in z with defining polynomial w^2 + i over its base field             sage: m.pari_relative_polynomial()             Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1)              sage: l.<t> = m.extension(m['t'].0^2 + z)             sage: l.pari_relative_polynomial()             Mod(1, y^4 + 1)*x^2 + Mod(y, y^4 + 1) Return number of roots of unity in this relative field.          EXAMPLES::              sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] )             sage: K.number_of_roots_of_unity()             24 Return all the roots of unity in this relative field, primitive or not.          EXAMPLES::              sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] )             sage: rts = K.roots_of_unity()             sage: len(rts)             24             sage: all(u in rts for u in [b*a, -b^2*a - b^2, b^3, -a, b*a + b])             True Return the chosen generator over `\QQ` for this relative number field.          EXAMPLES::              sage: y = polygen(QQ,'y')             sage: k.<a> = NumberField([y^2 + 2, y^4 + 3])             sage: g = k.absolute_generator(); g             a0 - a1             sage: g.minpoly()             x^2 + 2*a1*x + a1^2 + 2             sage: g.absolute_minpoly()             x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49 Return ``self`` as an absolute number field.          INPUT:          - ``names`` -- string; name of generator of the absolute field          OUTPUT:          An absolute number field `K` that is isomorphic to this field.          Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where         ``from_K`` is an isomorphism from `K` to ``self`` and ``to_K``         is an isomorphism from ``self`` to `K`.          EXAMPLES::              sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K             Number Field in a with defining polynomial x^4 + 3 over its base field             sage: L.<xyz> = K.absolute_field(); L             Number Field in xyz with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49             sage: L.<c> = K.absolute_field(); L             Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49              sage: from_L, to_L = L.structure()             sage: from_L             Isomorphism map:               From: Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49               To:   Number Field in a with defining polynomial x^4 + 3 over its base field             sage: from_L(c)             a - b             sage: to_L             Isomorphism map:               From: Number Field in a with defining polynomial x^4 + 3 over its base field               To:   Number Field in c with defining polynomial x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49             sage: to_L(a)             -5/182*c^7 - 87/364*c^5 - 185/182*c^3 + 323/364*c             sage: to_L(b)             -5/182*c^7 - 87/364*c^5 - 185/182*c^3 - 41/364*c             sage: to_L(a)^4             -3             sage: to_L(b)^2             -2 Return defining polynomial of this number field         as a pair, an ntl polynomial and a denominator.          This is used mainly to implement some internal arithmetic.          EXAMPLES::              sage: NumberField(x^2 + (2/3)*x - 9/17,'a').absolute_polynomial_ntl()             ([-27 34 51], 51) Return the polynomial over `\QQ` that defines this field as an         extension of the rational numbers.          .. NOTE::              The absolute polynomial of a relative number field is             chosen to be equal to the defining polynomial of the             underlying PARI absolute number field (it cannot be             specified by the user).  In particular, it is always a             monic polynomial with integral coefficients.  On the other             hand, the defining polynomial of an absolute number field             and the relative polynomial of a relative number field are             in general different from their PARI counterparts.          EXAMPLES::              sage: k.<a, b> = NumberField([x^2 + 1, x^3 + x + 1]); k             Number Field in a with defining polynomial x^2 + 1 over its base field             sage: k.absolute_polynomial()             x^6 + 5*x^4 - 2*x^3 + 4*x^2 + 4*x + 1          An example comparing the various defining polynomials to their         PARI counterparts::              sage: k.<a, c> = NumberField([x^2 + 1/3, x^2 + 1/4])             sage: k.absolute_polynomial()             x^4 - x^2 + 1             sage: k.pari_polynomial()             x^4 - x^2 + 1             sage: k.base_field().absolute_polynomial()             x^2 + 1/4             sage: k.pari_absolute_base_polynomial()             y^2 + 1             sage: k.relative_polynomial()             x^2 + 1/3             sage: k.pari_relative_polynomial()             x^2 + Mod(y, y^2 + 1)*x - 1 Return the defining polynomial of this relative number field over its base field.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])             sage: K.relative_polynomial()             x^2 + x + 1          Use absolute polynomial for a polynomial that defines the absolute         extension.::              sage: K.absolute_polynomial()             x^6 + 3*x^5 + 8*x^4 + 9*x^3 + 7*x^2 + 6*x + 3 Return the defining polynomial of this relative number field.          This is exactly the same as ``relative_polynomial()``.          EXAMPLES::              sage: C.<z> = CyclotomicField(5)             sage: PC.<X> = C[]             sage: K.<a> = C.extension(X^2 + X + z); K             Number Field in a with defining polynomial X^2 + X + z over its base field             sage: K.defining_polynomial()             X^2 + X + z For a relative number field, ``polynomial()`` is deliberately         not implemented.  Either :meth:`~relative_polynomial` or         :meth:`~absolute_polynomial` must be used.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])             sage: K.polynomial()             Traceback (most recent call last):             ...             NotImplementedError: For a relative number field L you must use either L.relative_polynomial() or L.absolute_polynomial() as appropriate Return the base field of this relative number field.          EXAMPLES::              sage: k.<a> = NumberField([x^3 + x + 1])             sage: R.<z> = k[]             sage: L.<b> = NumberField(z^3 + a)             sage: L.base_field()             Number Field in a with defining polynomial x^3 + x + 1             sage: L.base_field() is k             True          This is very useful because the print representation of         a relative field doesn't describe the base field.::              sage: L             Number Field in b with defining polynomial z^3 + a over its base field This is exactly the same as base_field.          EXAMPLES::              sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1])             sage: k.base_ring()             Number Field in a1 with defining polynomial x^3 + x + 1             sage: k.base_field()             Number Field in a1 with defining polynomial x^3 + x + 1 Compute all field embeddings of the relative number field self         into the field `K` (which need not even be a number field,         e.g., it could be the complex numbers). This will return an         identical result when given `K` as input again.          If possible, the most natural embedding of self into `K`         is put first in the list.          INPUT:          - ``K`` -- a field          EXAMPLES::              sage: K.<a,b> = NumberField([x^3 - 2, x^2+1])             sage: f = K.embeddings(ComplexField(58)); f             [             Relative number field morphism:               From: Number Field in a with defining polynomial x^3 - 2 over its base field               To:   Complex Field with 58 bits of precision               Defn: a |--> -0.62996052494743676 - 1.0911236359717214*I                     b |--> -1.9428902930940239e-16 + 1.0000000000000000*I,             ...             Relative number field morphism:               From: Number Field in a with defining polynomial x^3 - 2 over its base field               To:   Complex Field with 58 bits of precision               Defn: a |--> 1.2599210498948731                     b |--> -0.99999999999999999*I             ]             sage: f[0](a)^3             2.0000000000000002 - 8.6389229103644993e-16*I             sage: f[0](b)^2             -1.0000000000000001 - 3.8857805861880480e-16*I             sage: f[0](a+b)             -0.62996052494743693 - 0.091123635971721295*I # this should be concordant with automorphisms # If there is an embedding that preserves variable names # then it is most natural, so we put it first. Compute all Galois automorphisms of self over the base field.  This is         different than computing the embeddings of self into self; there,         automorphisms that do not fix the base field are considered.          EXAMPLES::              sage: K.<a, b> = NumberField([x^2 + 10000, x^2 + x + 50]); K             Number Field in a with defining polynomial x^2 + 10000 over its base field             sage: K.automorphisms()             [             Relative number field endomorphism of Number Field in a with defining polynomial x^2 + 10000 over its base field               Defn: a |--> a                     b |--> b,             Relative number field endomorphism of Number Field in a with defining polynomial x^2 + 10000 over its base field               Defn: a |--> -a                     b |--> b             ]             sage: rho, tau = K.automorphisms()             sage: tau(a)             -a             sage: tau(b) == b             True              sage: L.<b, a> = NumberField([x^2 + x + 50, x^2 + 10000, ]); L             Number Field in b with defining polynomial x^2 + x + 50 over its base field             sage: L.automorphisms()             [             Relative number field endomorphism of Number Field in b with defining polynomial x^2 + x + 50 over its base field               Defn: b |--> b                     a |--> a,             Relative number field endomorphism of Number Field in b with defining polynomial x^2 + x + 50 over its base field               Defn: b |--> -b - 1                     a |--> a             ]             sage: rho, tau = L.automorphisms()             sage: tau(a) == a             True             sage: tau(b)             -b - 1              sage: PQ.<X> = QQ[]             sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])             sage: PF.<Y> = F[]             sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)             sage: K.automorphisms()             [             Relative number field endomorphism of Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field               Defn: c |--> c                     a |--> a                     b |--> b,             Relative number field endomorphism of Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base field               Defn: c |--> -c                     a |--> a                     b |--> b             ] # absolute automorphisms Return the collection of all infinite places of self.          By default, this returns the set of real places as         homomorphisms into RIF first, followed by a choice of one of         each pair of complex conjugate homomorphisms into CIF.          On the other hand, if prec is not None, we simply return places         into RealField(prec) and ComplexField(prec) (or RDF, CDF if         prec=53).          There is an optional flag all_complex, which defaults to False. If         all_complex is True, then the real embeddings are returned as         embeddings into CIF instead of RIF.          EXAMPLES::              sage: L.<b, c> = NumberFieldTower([x^2 - 5, x^3 + x + 3])             sage: L.places()             [Relative number field morphism:             From: Number Field in b with defining polynomial x^2 - 5 over its base field             To:   Real Field with 106 bits of precision             Defn: b |--> -2.236067977499789696409173668937             c |--> -1.213411662762229634132131377426,             Relative number field morphism:             From: Number Field in b with defining polynomial x^2 - 5 over its base field             To:   Real Field with 106 bits of precision             Defn: b |--> 2.236067977499789696411548005367             c |--> -1.213411662762229634130492421800,             Relative number field morphism:             From: Number Field in b with defining polynomial x^2 - 5 over its base field             To:   Complex Field with 53 bits of precision             Defn: b |--> -2.23606797749979 ...e-1...*I             c |--> 0.606705831381... - 1.45061224918844*I,             Relative number field morphism:             From: Number Field in b with defining polynomial x^2 - 5 over its base field             To:   Complex Field with 53 bits of precision             Defn: b |--> 2.23606797749979 - 4.44089209850063e-16*I             c |--> 0.606705831381115 - 1.45061224918844*I] Return the absolute different of this relative number field `L`, as an         ideal of `L`. To get the relative different of `L/K`, use         ``L.relative_different()``.          EXAMPLES::              sage: K.<i> = NumberField(x^2 + 1)             sage: t = K['t'].gen()             sage: L.<b> = K.extension(t^4 - i)             sage: L.absolute_different()             Fractional ideal (8) Return the relative different of this extension `L/K` as         an ideal of `L`.  If you want the absolute different of         `L/\QQ`, use ``L.absolute_different()``.          EXAMPLES::              sage: K.<i> = NumberField(x^2 + 1)             sage: PK.<t> = K[]             sage: L.<a> = K.extension(t^4  - i)             sage: L.relative_different()             Fractional ideal (4) The different, unqualified, of a relative number field is deliberately         not implemented, so that a user cannot mistake the absolute different         for the relative different, or vice versa.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])             sage: K.different()             Traceback (most recent call last):             ...             NotImplementedError: For a relative number field you must use relative_different or absolute_different as appropriate Return the absolute discriminant of this relative number field         or if ``v`` is specified, the determinant of the trace pairing         on the elements of the list ``v``.          INPUT:          - ``v`` (optional) -- list of element of this relative number field.          OUTPUT: Integer if ``v`` is omitted, and Rational otherwise.          EXAMPLES::              sage: K.<i> = NumberField(x^2 + 1)             sage: t = K['t'].gen()             sage: L.<b> = K.extension(t^4 - i)             sage: L.absolute_discriminant()             16777216             sage: L.absolute_discriminant([(b + i)^j for j in range(8)])             61911970349056 Return the relative discriminant of this extension `L/K` as an ideal of         `K`. If you want the (rational) discriminant of `L/\QQ`, use e.g.         ``L.absolute_discriminant()``.          EXAMPLES::              sage: K.<i> = NumberField(x^2 + 1)             sage: t = K['t'].gen()             sage: L.<b> = K.extension(t^4 - i)             sage: L.relative_discriminant()             Fractional ideal (256)             sage: PQ.<X> = QQ[]             sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])             sage: PF.<Y> = F[]             sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)             sage: K.relative_discriminant() == F.ideal(4*b)             True          TESTS:          Number fields defined by non-monic and non-integral         polynomials are supported (:trac:`252`)::              sage: K.<a> = NumberField(x^2 + 1/2)             sage: L.<b> = K.extension(x^2 - 1/2)             sage: L.relative_discriminant()             Fractional ideal (2) The discriminant, unqualified, of a relative number field is deliberately         not implemented, so that a user cannot mistake the absolute discriminant         for the relative discriminant, or vice versa.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])             sage: K.discriminant()             Traceback (most recent call last):             ...             NotImplementedError: For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate The discriminant, unqualified, of a relative number field is deliberately         not implemented, so that a user cannot mistake the absolute discriminant         for the relative discriminant, or vice versa.          EXAMPLES::              sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])             sage: K.disc()             Traceback (most recent call last):             ...             NotImplementedError: For a relative number field you must use relative_discriminant or absolute_discriminant as appropriate Return the order with given ring generators in the maximal         order of this number field.          INPUT:          - ``gens`` -- list of elements of self; if no generators are given, just           returns the cardinality of this number field (oo) for consistency.         - ``check_is_integral`` -- bool (default: True), whether to check that each           generator is integral.         - ``check_rank`` -- bool (default: True), whether to check that the ring           generated by gens is of full rank.         - ``allow_subfield`` -- bool (default: False), if True and the generators           do not generate an order, i.e., they generate a subring of smaller           rank, instead of raising an error, return an order in a smaller           number field.          The check_is_integral and check_rank inputs must be given as         explicit keyword arguments.          EXAMPLES::              sage: P.<a,b,c> = QQ[2^(1/2), 2^(1/3), 3^(1/2)]             sage: R = P.order([a,b,c]); R             Relative Order in Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field          The base ring of an order in a relative extension is still `\ZZ`.::              sage: R.base_ring()             Integer Ring          One must give enough generators to generate a ring of finite index         in the maximal order::              sage: P.order([a,b])             Traceback (most recent call last):             ...             ValueError: the rank of the span of gens is wrong Determine whether or not `L/K` is free (i.e. if `\mathcal{O}_L` is         a free `\mathcal{O}_K`-module).          INPUT:          - ``proof`` -- default: True          EXAMPLES::              sage: x = polygen(QQ)             sage: K.<a> = NumberField(x^2+6)             sage: x = polygen(K)             sage: L.<b> = K.extension(x^2 + 3)    ## extend by x^2+3             sage: L.is_free()             False Factorisation of univariate polynomials over relative number fields.          This is called by the ``factor`` method of univariate polynomials.          EXAMPLES::              sage: K.<i> = NumberField(x**2+1)             sage: L.<sqrt2> = K.extension(x*x-2)             sage: x = polygen(L,'x')             sage: factor(x**2+8)  # indirect doctest             (x + 2*i*sqrt2) * (x - 2*i*sqrt2) Lift an element of this extension into the base field if possible,         or raise a ValueError if it is not possible.          EXAMPLES::              sage: x = polygen(ZZ)             sage: K.<a> = NumberField(x^3 - 2)             sage: R.<y> = K[]             sage: L.<b> = K.extension(y^2 - a)             sage: L.lift_to_base(b^4)             a^2             sage: L.lift_to_base(b^6)             2             sage: L.lift_to_base(355/113)             355/113             sage: L.lift_to_base(b)             Traceback (most recent call last):             ...             ValueError: The element b is not in the base field          TESTS:          Number fields defined by non-monic and non-integral         polynomials are supported (:trac:`252`)::              sage: R.<x> = QQ[]             sage: K.<a> = NumberField(x^2 + 1/2)             sage: L.<b> = K.extension(x^2 - a/2)             sage: L.lift_to_base(b^2)             1/2*a # Convert the element to a PARI polynomial with t_POLMOD # coefficients representing elements of the base field. # Lift the coefficients and call simplify() to make PARI check # which variables really appear in the resulting polynomial # (otherwise we always have a polynomial in two variables even # though not all variables actually occur). # Special case: check whether the result is simply an integer or rational # Now we should have a polynomial in the variable y. # Otherwise we're not in the base field. Given an element in self or an embedding of a subfield into self,         return a relative number field `K` isomorphic to self that is relative         over the absolute field `\QQ(\alpha)` or the domain of `\alpha`, along         with isomorphisms from `K` to self and from self to `K`.          INPUT:          - ``alpha`` -- an element of self, or an embedding of a subfield into self         - ``names`` -- name of generator for output field `K`.          OUTPUT: `K` -- a relative number field          Also, ``K.structure()`` returns ``from_K`` and ``to_K``, where         ``from_K`` is an isomorphism from `K` to self and ``to_K`` is         an isomorphism from self to `K`.          EXAMPLES::              sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K             Number Field in a with defining polynomial x^4 + 3 over its base field             sage: L.<z,w> = K.relativize(a^2)             sage: z^2             z^2             sage: w^2             -3             sage: L             Number Field in z with defining polynomial x^4 + (-2*w + 4)*x^2 + 4*w + 1 over its base field             sage: L.base_field()             Number Field in w with defining polynomial x^2 + 3          Now suppose we have `K` below `L` below `M`::              sage: M = NumberField(x^8 + 2, 'a'); M             Number Field in a with defining polynomial x^8 + 2             sage: L, L_into_M, _ = M.subfields(4)[0]; L             Number Field in a0 with defining polynomial x^4 + 2             sage: K, K_into_L, _ = L.subfields(2)[0]; K             Number Field in a0_0 with defining polynomial x^2 + 2             sage: K_into_M = L_into_M * K_into_L              sage: L_over_K = L.relativize(K_into_L, 'c'); L_over_K             Number Field in c with defining polynomial x^2 + a0_0 over its base field             sage: L_over_K_to_L, L_to_L_over_K = L_over_K.structure()             sage: M_over_L_over_K = M.relativize(L_into_M * L_over_K_to_L, 'd'); M_over_L_over_K             Number Field in d with defining polynomial x^2 + c over its base field             sage: M_over_L_over_K.base_field() is L_over_K             True          Test relativizing a degree 6 field over its degree 2 and degree 3         subfields, using both an explicit element::              sage: K.<a> = NumberField(x^6 + 2); K             Number Field in a with defining polynomial x^6 + 2             sage: K2, K2_into_K, _ = K.subfields(2)[0]; K2             Number Field in a0 with defining polynomial x^2 + 2             sage: K3, K3_into_K, _ = K.subfields(3)[0]; K3             Number Field in a0 with defining polynomial x^3 - 2          Here we explicitly relativize over an element of K2 (not the         generator)::              sage: L = K.relativize(K3_into_K, 'b'); L             Number Field in b with defining polynomial x^2 + a0 over its base field             sage: L_to_K, K_to_L = L.structure()             sage: L_over_K2 = L.relativize(K_to_L(K2_into_K(K2.gen() + 1)), 'c'); L_over_K2             Number Field in c0 with defining polynomial x^3 - c1 + 1 over its base field             sage: L_over_K2.base_field()             Number Field in c1 with defining polynomial x^2 - 2*x + 3          Here we use a morphism to preserve the base field information::              sage: K2_into_L = K_to_L * K2_into_K             sage: L_over_K2 = L.relativize(K2_into_L, 'c'); L_over_K2             Number Field in c with defining polynomial x^3 - a0 over its base field             sage: L_over_K2.base_field() is K2             True # alpha is an embedding of a subfield into self; compose to get an # embedding of a subfield into the absolute field # alpha is an element coercible into self Returns an element of self with valuation 1 at the prime ideal P.          INPUT:           -  ``self`` - a number field          -  ``P`` - a prime ideal of self          -  ``others`` - either "positive" (default), in which            case the element will have non-negative valuation at all other            primes of self, or "negative", in which case the element will have            non-positive valuation at all other primes of self.           .. note::             When P is principal (e.g. always when self has class number            one) the result may or may not be a generator of P!          EXAMPLES::              sage: K.<a, b> = NumberField([x^2 + 23, x^2 - 3])             sage: P = K.prime_factors(5)[0]; P             Fractional ideal (5, 1/2*a + b - 5/2)             sage: u = K.uniformizer(P)             sage: u.valuation(P)             1             sage: (P, 1) in K.factor(u)             True Used for unpickling old pickles.      EXAMPLES::          sage: from sage.rings.number_field.number_field_rel import NumberField_relative_v1         sage: R.<x> = CyclotomicField(3)[]         sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a')         Number Field in a with defining polynomial x^2 + 7 over its base field # historical reasons only