Admissible groups over number fields
Deependra Singh


§ ABSTRACT 
Given a field K, one may ask which finite groups are 
Galois groups of field
extensions L/K such that L 
is a maximal subfield of a 
division algebra with center K.
This 
connection between inverse Galois theory 
and division algebras 
was first explored by Schacher
in the 1960s. In this manuscript we consider 
this problem when K
is a number field.
For the case when L/K is
assumed to be tamely ramified,
we give
a complete classification
of number fields for which
every solvable Sylow-metacyclic 
group is admissible, extending
J. Sonn's result for K =.
For the case when L/K is 
 allowed to be wildly ramified,
we give a characterization of 
admissible groups over several
classes of number fields, 
and partial results in other cases.


Fair Minimum Representation Clustering via Integer Programming A preliminary version of this paper appeared in CPAIOR 2024.
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§ INTRODUCTION



A central simple algebra over a field K
is a finite dimensional
associative K-algebra
such that its center is K
and it has no non-trivial two-sided
ideals. It is called a central 
division algebra if every non-zero
element is a unit
(for example, the algebra of quaternions over ).
The dimension of a central division algebra as
a K vector space
is always a square <cit.>,
and the square-root of
this dimension is called
its index.
If D is a central 
K-division algebra of index n
then a subfield L of D containing K

is maximal among all such subfields
if and only if its degree over K is n <cit.>.
Such a subfield is called a 
maximal subfield of D. For example,
the complex numbers are a maximal subfield 
of the division algebra of quaternions over .

Given a field K, the classical
inverse Galois problem asks
whether or not every finite group 
appears as the Galois group of 
some Galois extension of K.
With the terminology in the previous paragraph,
one can also ask the following question.

 
Which finite groups
G are Galois groups of field extensions L/K
such that L is a maximal subfield of a 
central division 
algebra over K?


Such a group G is called
admissible over K or K-admissible, 
and the field L is called K-adequate.
This connection between
inverse Galois theory and division algebras
was first explored by Schacher <cit.>.
In this paper we investigate this problem further,
and prove results about admissible groups
 over number fields.
See the next section for discussion concerning the 
motivation for this problem.


Over number fields, 
most results have focused on tamely
ramified adequate extensions and Sylow metacyclic
subgroups <cit.>, <cit.>
(Sylow-metacyclic 
groups are those whose Sylow subgroups are metacyclic).
Our results concern both
tamely and wildly ramified adequate
extensions. 
For tamely ramified adequate extensions,
we extend Sonn's result <cit.>, and
characterize number fields over which every
solvable Sylow-metacyclic group is 
tamely admissible
(see Definition <ref> for
the term “tamely admissible”).
More precisely, we show the following result
in Theorem <ref>:

    Let K be a number field. Then
    
        
  * A solvable Sylow-metacyclic group
        is tamely admissible over K if and only if
        each of its Sylow subgroups are tamely admissible
        over K.
        
  * Every 2-metacyclic group is tamely admissible
        over K if and only if 
    K ∩{√(-1), √(2), √(-2)} = ∅.

        
  * Let p be any odd prime, and let α_p
        be a primitive element of the unique
        degree p-extension over  contained in
        the field extension (ζ_p^2)/.
        Then every p-metacyclic group is tamely
        admissible over K if and only if α_p ∉ K.
    


While a -admissible group is
necessarily Sylow-metacyclic
(Theorem 4.1 of <cit.>),
it is also known that for any given finite group G
there is some number field K over which G is admissible
(Theorem 9.1 of <cit.>).
So a natural question is to understand how admissible groups
behave as we go to higher degree number fields.
As we will see, if the admissible group is not Sylow-metacyclic
then any corresponding
adequate extension must be wildly ramified.
We investigate
this phenomenon further,
and the following theorem describes a key result
for admissibility of p-groups in this context
(see Theorem <ref>):


    Let K be a finite Galois extension of , and p
    be an odd rational prime such that ζ_p ∉ K,
    and p decomposes in K. Let G be a p-group. Then
    
        
  * If ζ_p ∉ K_p
            then G is K-admissible
            if and only if d(G) ≤ [K_:_p] + 1.
        
  * If ζ_p ∈ K_p
            then G is K-admissible
            if and only if G 
            can be generated by [K_:_p] + 2
            many generators x_1, x_2, …, x_n
            satisfying the relation
            
    x_1^p^s[x_1,x_2][x_3,x_4] … [x_n-1,x_n] = 1

            where p^s is such that ζ_p^s∈ K_p 
            but ζ_p^s+1∉ K_p.
    

Here d(G) denotes the minimum number of 
generators of G.

The admissibility problem in the general case 
is open even in the case of p-groups.
The key challenge seems to be to handle
the case when ζ_p ∈ K.
But if we narrow our scope to special classes of
number fields, more can be said. 
The following result characterizes the
admissibility of odd p-groups
over quadratic number fields
(see Corollary <ref>
for this assertion, 
and Definition <ref>
for the term “decompose”):


	Let K be a quadratic number field,
	and G be an odd p-group for some rational prime p.
	Then G is K-admissible
	if and only if one of the following conditions
	holds:
	
		
  * prime p decomposes in K and d(G) ≤ 2, or,
		
  * prime p does not decompose in K and G is metacyclic.
	


Analogous results for number fields of 
degree 3 or 4 over  appear in Propositions 
<ref> and  <ref>.
We also discuss
results over Galois number
fields,
number fields of
degree 2^n
over ,
number fields
of odd degree over , 
and cyclotomic number fields.



This paper is organized as follows.
Section 1 provides additional
motivation and context for the admissibility problem.
In Section 2, we discuss 
admissibility of Sylow-metacyclic
group over number fields,
and characterize number fields
for which every solvable Sylow-metacyclic
subgroup is admissible,
extending Sonn's result.
Section 3 goes beyond
Sylow-metacyclic groups and
studies how degree of the number
field influences the class
of admissible groups.
Finally, in Section 4,
we specialize to special
classes of number fields 
where we can make stronger statements,
including Galois number fields,
cyclotomic number fields,
and number fields of degrees
2,3, and 4. 
We sometimes include extra hypotheses 
in stating results
where doing so would make the statements 
simpler,
and indicate how the results 
extend to more general situations.



 §.§ Acknowledgements

The author would like to thank Professors David Harbater,
Daniel Krashen, and Florian Pop for a number of 
very helpful conversations concerning material in
this manuscript and related ideas.
This paper is part of author’s Ph.D. thesis, 
completed under the supervision of 
Prof. David Harbater at the University of Pennsylvania.


§ BACKGROUND AND MOTIVATION


The following observations provide motivation
for studying the admissibility problem.

(i) Cross product algebras provide an explicit
way to 
work with central simple algebras over a field.
More specifically, each Brauer class α∈Br(K)
has a representative central simple algebra
which is a G-cross product algebra over K 
for some finite group G,
but a division algebra need not be a
cross product algebra in general.
On the other hand,
essentially by definition,
a finite group G is admissible over
(a  field) K if and only if there is
a G-cross product division algebra over K.
    
(ii) Let K be a field such that 
per(α) = ind(α)
for every Brauer class α∈Br(K)
(for example,
a global field or a local field).
Let L/K be a finite G-Galois extension
with n = [L:K].
Then L is K-adequate if and only
if the n-torsion abelian group
H^2(G, L^⋆) has an element
of order exactly equal to n
(Proposition 2.1 of <cit.>).

(iii) Let K be a field and let f(x) ∈ K[x]
be an irreducible polynomial.
One may ask whether there exists a
(finite dimensional) central division
algebra over K containing a root α of f(x).
If 
per(α) = ind(α)
for every α∈Br(K),
then such a division algebra exists
if and only if the Galois closure
of K(α) is a K-adequate extension
(follows from Proposition 2.2 of <cit.>).
This was Schacher's motivation in the original paper
to study the admissibility problem.

    

In light of Question <ref>,
the admissibility problem can be thought of
as a non-commutative version of the inverse 
Galois problem.
In particular,
a K-admissible finite group first needs
to be a Galois group over K.
Thus if 
K has no non-trivial Galois
groups (e.g., K is separably
closed), then no
non-trivial group is admissible over K.
Similarly, if Br(K) = 0
then there are
no non-trivial admissible groups
over K since there are no
non-trivial K-central division algebras.
This is true in particular for
C_1 fields (quasi-algebraically closed fields),
which includes the following fields:

    
  * separably closed fields;
    
  * finite fields;
    
  * function field of a smooth curve over
    an algebraically closed field, e.g., (t);
    
  * a complete discretely valued field
    with an algebraically closed residue field,
    e.g., ((t));
    
  * maximal
    unramified extension of a complete
    discretely valued field with a perfect 
    residue field, e.g., _p^ur.


Every finite group
is known to be
Galois over fields of type
(iii) 
in the above list
(by <cit.> in characteristic 0,
and <cit.> in 
characteristic p>0).
This shows that even
if every finite group is Galois 
over a field K, there
may not be any non-trivial groups
admissible over K.
On the other hand, if K is a local field,
then every finite group
which is Galois over K is
also admissible over K.
In fact, the following
stronger statement is true.


    If K is a local field,
    then every finite Galois extension L/K 
    is K-adequate.

To see this, note that since period
equals index for local fields,
L is K-adequate if and only
if H^2(G, L^⋆) has an element
of order [L:K] 
(Proposition 2.1 of <cit.>).
But H^2(G, L^⋆) is cyclic
of order [L:K] for a local field K,
and the result follows.



Like the inverse Galois problem, 
the admissibility problem remains open
in general. 
But unlike the inverse Galois problem, the
 groups that occur in this fashion are often
 quite restricted. 
For example, while every finite 
group is expected to be realized as a Galois group over ,
by Theorem 4.1 of <cit.>
a -admissible group must be Sylow-metacyclic
(a metacyclic group is an extension
of a cyclic group by another cyclic group). 
On the other hand,
every finite group is 
admissible over some 
number field (Theorem 9.1 of <cit.>).


While the problem is open in general, 
including over ,
some results are known. 
Sonn <cit.> proved the admissibility
of solvable Sylow metacyclic groups over .
Many non-solvable groups 
with metacyclic Sylow subgroups 
have also been shown to be
admissible over  as well 
as over other classes of number fields,
for example <cit.>, <cit.>, 
<cit.>, <cit.>.
Since not every non-solvable 
Sylow metacyclic group is
known to be even Galois over  <cit.>, 
the problem of completely
characterizing admissible groups 
over number fields remains
out of reach at present. 
In <cit.>, groups that are 
admissible over function fields over
certain complete discretely valued 
fields were characterized using patching 
techniques. 




Since the Brauer group is intimately
related to division
algebras over a field, it
plays a key role in studying 
admissibility.
Let K be a global field,
and L/K be a G-Galois
extension for some finite group G.
For a place  of K,
let D_ denote
the decomposition group
corresponding to 
for the field extension L/K,
and let L_
denote a completion
of L with respect to
an extension of 
the absolute value of K 
corresponding to .
By Proposition 2.1 of <cit.>,
L is K-adequate if and only
if H^2(G, L^⋆) has an element
of order exactly equal to [L:K].
Using this observation and the exact sequence 

    0 → H^2(G, L^*) →⊕_ H^2(D_, L_^*) →/→ 0

from class field theory, Schacher <cit.>
obtained the following arithmetic criterion 
for the extension
L/K to be K-adequate:

[Schacher's Criterion] 
Let K be a global field,
and L/K be a G-Galois
extension for some finite group G.
The field extension L/K is 
K-adequate if and only if for 
each rational prime p dividing the order of G,
there are two distinct places _1,_2 of K
such that the decomposition groups 
corresponding to these places 
in the field extension L/K contain
a p-Sylow subgroup of G.


According to Schacher's criterion,
admissibility of G over
K is equivalent to the existence
of a G-Galois
field extension of K 
(“inverse Galois problem”) with 
certain conditions
on the decomposition groups
of places of K (“local conditions”).
This refinement of the
inverse Galois problem with 
extra local conditions
is a problem that
is open in general, including for solvable groups.
For example, while Shafarevich's construction
shows that every solvable group can be 
realized as a Galois group over any number field,
there is no known way to realize the given local
extensions <cit.>.
Grunwald-Wang theorem (Theorem 5 of Chapter 10
in <cit.>) was the first result
of this kind for cyclic Galois
extensions,
and the most far 
reaching result of this kind is 
Neukirch's generalization of the
Grunwald-Wang theorem to solvable
groups of order coprime to roots
of unity in the global field
(Theorem 9.5.9 of <cit.>).
We make extensive use of this result
in addition to results on embedding
problems.

Observe that if the group G is a p-group
for some rational prime p then 
in Schacher's criterion above 
the decomposition groups corresponding
to places _1, _2 need to be
the whole group G. 
In this sense, the structure of the 
Galois group of the maximal p-extension
of a local field yields
important insights into
the admissibility problem.


§ SYLOW METACYCLIC GROUPS


We start with two definitions.


    Let G be a group. We say
    G is a metacyclic group if it is an extension
    of a cyclic group by another
    cyclic group. I.e., there
    is an exact sequence of groups:
    
    1 →/n→ G →/m→ 1

    for some integers m,n > 1.
    

 
    A Sylow-metacyclic group is
    a group such that all of its
    Sylow subgroups are metacyclic.



Schacher 
 observed that 
if K is a number 
field to which the p-adic 
valuation on  extends uniquely, 
then
the p-Sylow subgroups of any K-admissible group 
are necessarily metacyclic
(see Theorem 10.2 of <cit.>).
This follows immediately
from Schacher's criterion noted above.
In particular, this is true for the field of 
rational numbers , and 
so a -admissible group must be Sylow-metacyclic.

In the converse direction, Sonn <cit.>
proved that every solvable Sylow-metacyclic
group is admissible over .
As noted before, there are examples
of non-solvable Sylow metacyclic groups
that are not even known to be Galois
over , so the converse direction
remains open for non-solvable groups.

With this background,
one may ask whether every 
Sylow-metacyclic group is admissible
over every number field.
This is known to be false.
For example, the dihedral
group of order 8 is not admissible over
(i) (see <cit.>).
So we can refine our question
and ask:


 
    Can we classify the number fields K for which every
solvable Sylow-metacyclic group is
 K-admissible?


In this direction, 
Liedahl
proved a necessary 
and sufficient criterion for 
a given 
odd metacyclic p-group to be 
admissible over a given number field (Theorem 30 of <cit.>). 
Before we state this result,
we set some notation
 and terminology.

 
For a number field K/, 
    we say that a rational prime p
    decomposes in K if the
    p-adic valuation on  extends 
    to at least two inequivalent valuations on K.



 
   We say that a group G is tamely admissible over K 
   if there exists
   an adequate G-Galois extension L/K that is tamely ramified over K.



    Let ζ_e ∈ be a primitive
    e-th root of unity for
    some integer e ≥ 1.
    For an integer
    q coprime to e, let σ_e,q
    denote the field automorphism
    of (ζ_e)/
    determined by
    σ_e,q(ζ_e)
    =ζ_e^q.


 
    Let K be a number field
    and let G be a metacyclic
    p-group for some prime
    number p. We
    say that G admits
    a Liedahl presentation
    for K if G 
    admits a presentation
    
    ⟨ x,y  | 
        x^e = 1, y^f = x^i, 
        yxy^-1 = y^q
        ⟩

    such that (ζ_e) ∩ K
    ⊆(ζ_e)
    is contained in the fixed field of σ_e,q.
    
    K(ζ_e) [dl, dash] [dr, dash]   
    (ζ_e) [dr, dash]        K [dl, dash] 
       (ζ_e) ∩ K[d, dash]

    


With this notation, we can restate Liedahl's result as
follows.
 
Let K be a number field,
and let G be an odd metacyclic
p-group for some prime number 
p. Then G is admissible
over K if and only if at least
one of the following holds:

    
  * p decomposes
    in K.
    
  * G has a Liedahl
    presentation for K.

    



This criterion was later 
extended 
by Neftin 
to solvable 
Sylow metacyclic groups 
under the assumption that the 
adequate extension is 
tamely ramified
(Theorem 1.3 of <cit.>).
With our terminology, we can restate Neftin's result
as follows.
 
 
    Let K be a number field
    and G be a solvable
    group.
    Then 
    G is tamely admissible
    over K
    if and only if for each
    prime p dividing the order
    of the group G, 
    the p-Sylow subgroups
    of G admit a Liedahl
    presentation for K.


Note that this result has
an extra hypothesis of
tame admissibility.
Building on their work,
we classify number fields
for which all solvable
Sylow-metacyclic groups
are admissible.
This generalizes Sonn's result
that  is such a number
field,
and provides a complete 
answer to 
Question <ref>.
This is the main result
of this section, and is the content
of Theorem <ref>.


We first need some auxiliary
lemmas.
The following lemma is a well-known result and
follows from a group theory argument. 
We include here
a proof for completeness.
 
    Every metacyclic group G is a quotient
    of a semidirect product G' of two cyclic groups.
    Moreover, if G is a p-group for some prime
    number p, then G' can be chosen to be a p-group.



    Let G be a metacyclic group
    with presentation 
    
    ⟨ x,y | x^e = 1, y^f =x^i, yxy^-1 = x^q ⟩.

    Let r be the order of y in G.
    Since x = y^rxy^-r = x^q^r,
    we have q^r ≡ 1 (  e).
    This allows us to define the semidirect product
    G' = /e⋊/r with presentation
    
    ⟨x̃,ỹ|x̃^e = 1, 
        ỹ^r = 1, ỹx̃ỹ^-1 = x̃^q ⟩.

    Mapping x̃→ x, ỹ→ y
    defines a surjective group homomorphism
     G' ↠ G.

     It is clear from the construction that
     if G is a p-group for 
     some rational prime p
     then so is G'.


 
    Let k be a 
    non-archimedean
    local field,
    and k'/k be a tamely
    ramified G-Galois extension. 
    Let e be the ramification
     index and f
     be the residue
     degree of this extension,
     and let q be the number of elements in the 
     residue field of k.
    Then G has a presentation
    
    <x,y   |   x^e = 1, y^f = x^i, yxy^-1 = x^q>

    for an appropriate integer 
    i. In particular,
     G is metacyclic.




The following result is a consequence of Proposition
2.2 of <cit.>.
 
    Let K be a field such that (α) = (α)
    for every α∈(K)
    (e.g., a global field). 
    If G is admissible (resp., tamely admissible) over
    K and N ⊴ G is a normal subgroup then G/N
    is  admissible (resp., tamely admissible) over K.



The following lemma shows 
that the presence of roots
of unity constrains the tamely ramified
admissible groups to be “more abelian”.

 
    Let k be a non-archimedean local field and l
    a prime different from the residue
    characteristic of k.
    If ζ_l^n∈ k
    for some n ≥ 0,
    then any Galois l-extension
    of degree dividing l^n+1 is necessarily
    abelian.





    Let k'   |  k be a G-Galois
    extension of degree d such that d | l^n+1.
    Let e be the ramification index and 
    f be the residue degree
    of this extension.
    
    Since the residue characteristic of
    k is different from l, the extension
    k'   |   k is tamely ramified.
    If e = l^n+1 then the extension k' | k
    is totally and tamely ramified, and therefore
    it is a cyclic extension (Corollary 1 to 
    Proposition 4.1 in <cit.>).
    So without loss of generality
    we can assume that e   |   l^n.

    Let m | k be the maximal unramified extension
    inside k' | k. 

       
    k' [d, dash, "e"]  
    
        m [d, dash, "f"] 
    
        k

    
    Then k' | m and m | k
    are cyclic Galois extension of degrees e and f
    respectively. By Galois theory, there is an
     exact sequence of groups:
     
    1 →/e→ G →/f→ 1.


     

     Here G is the Galois group
     of k'   |   k. 
     Moreover, G has a presentation (see Lemma <ref>):
     
    <x,y   |   x^e = 1, y^f = x^i, yxy^-1 = x^q>

     where q is the number of elements in the 
     residue field of k.

     Since char(k) ≠ l and ζ_l^n∈ k,
     we must have ζ_l^n∈_q (the residue
     field).
     Therefore the order of
     subgroup ⟨ζ_l^n⟩⊆_q^*
     divides q-1,
     the order of _q^*.
     Since e divides
     l^n, 
     it follows that e
     divides q-1.
     Therefore x^q = x in G,
     and thus yxy^-1 = x in the above presentation.
     This implies
     that G is abelian.




Note that n needs to be at least 2 for the above lemma
to say something non-trivial since a group of
order l or l^2 is necessarily abelian.


 
    Let K be an number field
    and p be a prime number
    such that ζ_p^n∈ K
    and p does not decompose in K.
    Let G be a finite group such that its
    p-Sylow subgroup is non-abelian 
    of order ≤ p^n+1.
    Then G is not admissible 
    over K.


    If G were admissible over K,
    then by Schacher's Criterion 
    (see Criterion <ref>) there 
    will be two distinct places 
    _1, _2 of K
    such that K__1 and K__2
    admit Galois
    extensions with Galois groups
    containing a p-Sylow subgroup of G.

    Since p ∈ does not decompose in 
    K, one of these two places
    must have residue characteristic different
    from p. Without loss of generality,
    assume that _1 is that place,
    and let k = K__1. 
    
    Let k'   |   k
    be a Galois extension of local fields
    such that the Galois group contains a 
    p-Sylow subgroup H of G.
    Let m be the fixed field of H in this extension,
    and so k'   |   m is a H-Galois
    extension.

            
    k' [d, dash, "H"]  
    
        m [d, dash] 
    
        k

    
    Since ζ_p^n∈ K ⊂ k ⊆ m,
    and residue characteristic of m is different from
    p, this contradicts Lemma <ref> 
    since H is non-abelian.



 
    Let p be a rational prime number.
    Then the unique non-abelian group /p^2⋊/p
    is not admissible over (ζ_p^n) for n ≥ 2.



    Since p does not
    decompose in (ζ_p^n)
    and the p-Sylow subgroup 
    is the whole group,
    the result 
    follows from the previous corollary.




    This generalizes the observation in <cit.>
    that the dihedral group D_8 of order 8 is
    not admissible over Q(i).
    For example, the unique non-abelian
    group /9 ⋊/3 
    is not admissible over (ζ_9).


In fact, there is another
field strictly contained
in (ζ_p^2)
over which 
the non-abelian group /p^2⋊/p is not admissible.
This is shown in Lemma <ref>
and requires the following 
lemma as an ingredient.


 
    Let k'/k be a finite G-Galois extension
    of non-archimedean
    local fields where G is a p-group
    for a prime number p different
    from the residue characteristic of k.
    If the ramification index is p
    then G is abelian.



    Let m be the maximal unramified extension of k
    contained in k'.
    By Lemma <ref>,
    G sits in an exact sequence
    
    1 →Gal(k'/m)=/p→ G →Gal(m/k)=/p^a→ 1,

    and has a presentation 
    
    ⟨ x, y | x^p = 1, y^p^a = x^i, yxy^-1=x^q ⟩

    for some appropriate i, and q is the number of 
    elements in the residue
    field of k.
    Here x generates the inertia group,
    and y is a lift of the Frobenius automorphism.
    
    Since q and p are coprime,
    we have y^p-1xy^p-1 = x^q^p-1 = x,
    i.e., y^p-1 and x
    commute with each other. 
    But y' = y^p-1 is another lift of the Frobenius automorphism.
    So x and y' generate G, and therefore G is abelian.


 
    For any rational prime number l, the non-abelian
    semi-direct product /l^2 ⋊/l
    is not admissible over the unique
    degree l number field K
    inside (ζ_l^2).



    If G = /l^2 ⋊/l were admissible
    over K, then by Schacher's Criterion (see Criterion <ref>)
    there would exist a G-Galois
    extension L/K such that over two places
    of K, the decomposition
    group would be the whole group G.
    We show that this is not possible.
    
    Since l totally ramifies in K,
    one of these places must have residue
    characteristic different from l.
    Let  be that place of K,
    and p be its residue characteristic.
    Let k = K_,
    and k'/k be a 
    G-Galois extension.
    Note that since p ≠ l,
    the rational prime p is unramified in
    K/.
    This is because
    K ⊂(ζ_l^2)
    and the only prime
    that ramifies in (ζ_l^2)/ is l.

    Since G is non-abelian, 
    must be non-archimedean.
    Since the residue characteristic of
     is different from l,
    k'/k is a tamely ramified extension.
    Since G is non-abelian, the 
    ramification index cannot be l
    by Lemma <ref>.
    As a result, the only possibility
    for the ramification index is l^2.

    Let m be the maximal unramified extension
    inside k'/k,
    and so k'/m is totally and tamely ramified extension
    of degree l^2.
    Therefore ζ_l^2∈ m.
    If q is the number of elements in the
    residue field of k,
    then q^l ≡ 1  (  l^2).

            
    k' [d, dash, "l^2"]  
    
        m [d, dash, "l"] 
    
        k [d, dash, "f"]
    _p

    

    We now look at the splitting behavior of 
    rational prime p ≠ l in the extension
    K/. Since K/ is an abelian extension,
    this is determined by class field theory.

    First consider the case
    when prime p splits in K.
    This happens
    if and only if the order f of p 
    in (/l^2)^* divides
    (l-1). 
    If p ≡ 1  (  l^2),
    then k already has ζ_l^2,
    and so by Lemma <ref>,
    this is not possible.
    Otherwise, p^l ≢1  (  l^2)
    and since p = q, we get
    ζ_l^2∉ m. So this case is not possible either.

    Finally, consider the case when prime p stays inert 
    in K. In this case q = p^l (since [K:] = l), 
    and so we have q^l = p^l^2≡ 1  (  l^2).
    But p^l(l-1)≡ 1  (  l^2),
    and therefore p^l  ≡ 1  (  l^2).
    But this means that q  ≡ 1  (  l^2),
    and hence ζ_l^2∈ k.
    But this contradicts Lemma <ref>.



The following
lemma isolates a useful result
whose main ideas are
contained in the proof of
Theorem 27 and 28 of
<cit.>.

 
    Let K be a number field
    and G be a metacyclic 
    p-group for some prime 
    number p. Then the following
    are equivalent.
    
    
  * G is tamely admissible
    over K.
    
  * There are infinitely
    many rational primes l
    such that l splits completely
    in K and 
    _l admits
    a G-Galois extension.
    
  * There is a non-archimedean
    place v of K
    with residue characteristic
    different from p
    such that the completion K_v
    admits a G-Galois extension.
    


    (i)(ii).
    Since G is a solvable
    Sylow-metacyclic group
    and it is tamely admissible
    over K, the hypotheses of
    Theorem 1.3
    of <cit.> (or see
    its reformulation
    as Theorem <ref>
    above)
    are satisfied.
    Therefore  
    G has a 
    Liedahl presentation
    for K
    (see Definition <ref>
    for a definition of Liedahl presentation).
    
    It follows
    from the proof of 
    Theorem 27
    of <cit.>
    that if G has a Liedahl presentation for K
    then there are infinitely
    many rational primes l that completely
    split in K, 
    and have the property that
    _l admits a G-Galois extension.
    
    (ii)(iii) is clear.

    (iii)(i).
    Note that G is a solvable
    Sylow-metacyclic group.
    If we can show that
    G has a Liedahl
    presentation for K,
    then the tame admissibility
    of G over K will 
    follow from
    Theorem 1.3 of <cit.>.

    We now follow the proof of
    Theorem 28 in <cit.>
    to argue that G has a
    Liedahl presentation for K.
    Let k = K_v,
    and let k'/k
    be a G-Galois extension
    given in the hypothesis.
    
    Since the residue characteristic of k is
    different from p, and
    G is a p-group,
    it follows that k'/k
    is tamely ramified.
    Therefore, by Lemma <ref>,
    G has a presentation
    
    <x,y   |   x^e = 1, y^f = x^i, yxy^-1 = x^q>,

    where e is the ramification
    index, f is the residue 
    degree, and q is the number
    of elements in the residue 
    field of k.
    Since G is a p-group,
    both e and f are powers
    of p.

    Let ζ_e be a
    primitive e-th root
    of unity.
    In order to show that
    the presentation above is
    a Liedahl presentation of
    G for K, we need
    to argue that ζ_e ↦ζ_e^q
    fixes K ∩(ζ_e).
    By Galois theory, that is the
    same thing as ζ_e ↦ζ_e^q being 
    an automorphism of 
    K(ζ_e)/K.
    Since e is coprime to
    the residue
    characteristic
    of k, the
    extension k(ζ_e)/k
    is unramified. 
    Therefore ζ_e ↦ζ_e^q is
    an automorphism of 
    k(ζ_e)/k.
    Restricting
    k to K shows that 
    ζ_e ↦ζ_e^q is 
    an automorphism of 
    K(ζ_e)/K.



We are now in a position
to classify the number fields
for which every metacyclic p-
group is tamely admissible.
Starting with odd primes, we have the following

 
    Let K be a number field,
    and p be an odd rational prime.
    The following are equivalent:
    
        
  * Every metacyclic p-group 
        is tamely admissible over K.
        
  * The (unique) non-abelian
        group /p^2⋊/p
        is tamely admissible over K.
        
  * Let (α) be the 
        unique degree p subfield 
        of (ζ_p^2) for some
        primitive element α.
        Then α∉ K
        (equivalently, K ∩(ζ_p^2) ⊆(ζ_p)).
    



    (i)(ii) is clear
    since G = /p^2⋊/p is metacyclic.
    
    (ii)(iii).
    Let α be a primitive element as in the assertion (iii).
    For the sake of contradiction, assume that α∈ K.
    By Lemma <ref>,
    there exists a 
    rational prime l
    that splits completely in K,
    such that the local field _l
    admits G = /p^2⋊/p
    as a Galois group.
    Since (α) ⊂ K,
    the prime l must split in 
    (α) as well, and therefore
    by Lemma <ref> (iii),
    G is admissible over (α).
    This contradicts Lemma <ref>.
    Therefore, α∉ K.

    (iii)(i).
    Assume that α∉ K for a primitive
    element as in (iii).
    By Lemma <ref>
    and <ref>,
    it suffices to show that every semidirect
    product of cyclic p-groups
    is admissible over K.
    So let 
    G = /e⋊/f

    be a semi-direct product of cyclic p-groups
    with a presentation
    
    ⟨ x,y | x^e = 1, y^f =1, yxy^-1 = x^q ⟩,

    
    corresponding to a group action φ
    
    φ :  /f→Aut(/e).


    

    Since  Aut(/e)
    ≅Gal((ζ_e)/),
    the action can also be thought of as
    
    φ̃ :  /f→Gal((ζ_e)/),

    where the generator
    of the group /f
    maps to
    the automorphism ζ_e ↦ζ_e^q.
    
    Let H = im(φ̃), and consider the following diagram of field extensions.
    
    K(ζ_e) [dl, dash] [dr, dash]   
    (ζ_e) [d, dash] [dr, dash]        K [dl, dash] 
    
        M =(ζ_e)^H [dr, dash]     L = (ζ_e) ∩ K[d, dash]


    If we can show that the
    above presentation for G is
    a Liedahl presentation
    for K, 
    then
     Theorem 1.3 of <cit.> (or its
    reformulation
    as in Theorem <ref>)
    shows that G is tamely admissible
    over K. 
    To show that the above presentation
    (4.1) is a Liedahl
    presentation, we need to
    argue that L is
    fixed by the automorphism
    
    σ_e,q = ζ_e ↦ζ_e^q
         ∈Aut((ζ_e)/).

    The automorphism σ_e,q
    is a generator of
    H, and the fixed field of
    σ_e,q is M.
    Thus it suffices
    to show that L ⊆ M.
We do this using basic Galois theory.

    The extension (ζ_e)/ is an extension
    of degree (p-1)p^i for some i ∈∪{0}.
    Since H is a p-group (being the image of a p-group),
    its fixed field M must have degree (p-1)p^j
    over  (for some j ≤ i).


    On the other hand, 
    since (ζ_e)/ is a cyclic
    extension and
    the extension (α)/ is of
    degree p, (α) ⊆ L
    if and only if p | [L:].
    Since we are assuming that
    α∉ L,
    we must have that p ∤ [L:].
    Equivalently, [L:] | (p-1).
    This means [L:] | [M:].
    Once again, since (ζ_e)/ is a cyclic
    extension, this implies L ⊆ M
    and we are done.



For the even prime 2, the situation 
is a bit more involved,
and we need to consider 
degree two extensions in 
(ζ_8).
To formulate the precise
result, we recall
some notation.
Let Q_16
be the generalized
quaternion group of
order 16 with presentation

    ⟨ x,y | x^8 = 1, x^4 = y^2, yxy^-1 = x^7⟩.

Let SD_16
be the semi-dihedral group 
of order 16 with presentation

    ⟨ x,y | x^8 = 1 = y^2, yxy^-1 = x^3 ⟩.

We also need the following 
two lemmas.

 
    Let K be a number field such that K ∩{√(-1), √(-2)}≠∅.
    Then Q_16 is not tamely admissible over K.
    Moreover, if the 2-adic valuation on 
    extends uniquely to K, then Q_16
    is not admissible over K
    (either tamely or wildly).


    Using Schacher's Criterion <ref>, 
    it suffices to show
    that there are no
    places  of K with residue characteristic
    different from 2 such that
    the completion
    k = K_ admits Q_16 as a Galois group.

    Let k
    be such a completion,
    and let _q
    be the residue field of k.
    
    If √(-1)∈ K
    then q ≡ 1  (mod 4). 
    In this case, Theorem 3.1 of <cit.>
    says that k
    cannot admit
Q_16
    as a Galois group.

    If √(-2)∈ K
    then q ≡ 1  or  3  (mod 8).
    Once again,
    Theorem 3.1 of <cit.>
    asserts that
    k does not
    have a Q_16-Galois extension.

    Therefore, in both
    cases, k does not have a Q_16-Galois extension. 
    That is what
    we needed to
    conclude the 
    proof.


 
    Let K be a number field with √(2)∈ K.
    Then SD_16 is not tamely admissible over K.
    Moreover, if the 2-adic valuation on 
    extends uniquely to K, then SD_16
    is not admissible over K
    (either tamely or wildly).



    Using Schacher's Criterion <ref>, 
    it suffices to show
    both cases there are no
    places  of K with residue characteristic
    different from 2 such that k = K_ admits
    G = SD_16 as a Galois group.
    

    Suppose there were a G-Galois
    extension l/k with 
    ramification index e and residue
    index f.
    Of course e ≠ 1, 16
    since G is not a cyclic group.
    By Lemma <ref>,
    e cannot be 2 either
    since
    G is non-abelian.
    The group G has a unique 
    cyclic normal subgroup of order 4,
    but the quotient group is the Klein four-group, which
    is not cyclic. Thus e = 4 is not possible either.

    Therefore
    we must have
    e = 8 and 
    f = 2.
    Let _q be the residue field of
    k = K_. Since √(2)∈ K,
    q ≡± 1  mod 8.
    If q ≡ 1  mod 8 then
    ζ_8 ∈ k, which contradicts
    Lemma <ref>.
    Therefore  we
    must have q ≡ -1  mod 8.

    In this case, by Lemma <ref>, G
    has a presentation
    
    ⟨ a,b | a^8 = 1, b^2 = a^i,
        bab^-1 = a^q⟩

    for some integer i.
    Since q ≡ -1  mod 8,
    and a has order 8,
    this is the same
    thing as
    
    ⟨ a,b | a^8 = 1, b^2 = a^i,
        bab^-1 = a^-1⟩.

    But a quick check shows that 
    G has no two elements g,h
    such that g^8 = 1, and hgh^-1 = g^-1.



 
    For a number field K,
    the following are equivalent:
    
        
  * Every metacyclic 2-group 
        is tamely admissible over K.
        
  * The groups Q_16 and
SD_16 are tamely admissible over K.
        
  * K ∩{√(-1), √(2), √(-2)} = ∅
        (equivalently, K ∩(ζ_8) =).
    



    (i)(ii). This follows because
    both Q_16 and SD_16 are metacyclic.
    Both of them have a cyclic normal
    subgroup of order 8 and the quotient
    by that subgroup is the cyclic group of order 2
    (In fact, SD_16 is a semidirect product
    /8⋊/2).

    (ii)(iii).
    By Lemma <ref>,
    the number field K cannot contain either √(-1) or √(-2)
    since Q_16 is  tamely
    admissible
    over K.
    Similarly,
    by Lemma <ref>, K
    cannot contain √(2) since SD_16
    is tamely
    admissible over K.

    (iii)(i).
    The argument for this implication proceeds
    the same way as in the implication
    (iii)  (i) in Proposition
    <ref>.
    So let K,M,L as in that proof,
    and e is a power of 2.
    If L ≠ then L ∩{√(-1),  √(2),  √(-2)}≠∅
    since (i), (√(2)), 
    and (√(-2)) are the only 
    degree 2 extensions
    inside (ζ_e)
    (here, e is a power of 2).
    But
    K ∩{i, √(2), √(-2)}
    = ∅
    by hypothesis.
     Therefore L ∩{i, √(2), √(-2)}
    = ∅, and we get 
    L = ⊆ M.




By combining Proposition
<ref>
and Proposition
<ref>
we get the main
result of this section
as the following theorem.
Note that part (i)
of the theorem
reduces the admissibility
of a general solvable
Sylow-metacyclic group
to p-groups,
and those cases are handled
by Proposition 
<ref>
    and 
    Proposition <ref>.


 
    Let K be a number field. Then
    
        
  * A solvable Sylow-metacyclic group
        is tamely admissible over K if and only if
        all of its Sylow subgroups are tamely admissible
        over K.
        
  * Every 2-metacyclic group is tamely admissible
        over K if and only if 
    K ∩{√(-1), √(2), √(-2)} = ∅.

        
  * Let p be any odd prime, and let α_p
        be a primitive element of the unique
        degree p-extension over  contained in
        the field extension (ζ_p^2)/..
        Then every p-metacyclic group is tamely
        admissible over K if and only if α_p ∉ K.
    



    Part (i) follows from
    Theorem 1.3
    of <cit.>.
Part (ii) is Proposition <ref>,
    and 
    part (iii) is 
    Proposition <ref>.


We now describe some applications
of this theorem.


 
    Let K be a number field and G be a 
    metacyclic p-group
    for some prime number p. 
    If p is 
    tamely ramified in K/,
    then G is tamely admissible over K.
 
    

    If p is an odd prime 
    then it
    is wildly ramified in 
    (α_p)/,
    where α_p is as 
    defined in Theorem
    <ref>.
    Since p is assumed
    to be tamely ramified
    in K/, 
    it follows that 
    α_p ∉ K.
    A similar argument
    shows that {i, √(2), √(-2)}∩ K = ∅
    if p = 2.
    So the conclusion
    follows from Theorem <ref>.




 
    Let K be a number field,
    and let G be a solvable Sylow-metacyclic
    group with the property
    that for every prime
    number p that divides
    |G|, the prime  
    p is tamely ramified in K/.
    Then G is tamely admissible over K.



    By the previous corollary
    each p-Sylow subgroup 
    of G is tamely admissible
    over K. The result
    then follows from 
    Part (i) of Theorem <ref>. 




We note a characterization of admissible
solvable Sylow-metacyclic groups
in certain cases (with no restriction
on the adequate extension
being tamely ramified) as a corollary of Theorem <ref>:

 
    Let K be a number field,
    and let G be a solvable Sylow-metacyclic
    group with the property
    that if p divides |G|
    then p is tamely ramified
    in K/ and the p-adic valuation 
    on  extends uniquely to K.
    Then G is K-admissible if and only if G is Sylow-metacyclic.



    Let G be a K-admissible group.
    Let p be a rational prime that divides |G|.
    Since the p-adic
    valuation extends uniquely to K, 
    the p-Sylow subgroup of G 
    must be metacyclic by Theorem 10.2 of <cit.>.
    The converse direction follows from Theorem <ref>.



 
    Let K be an abelian number field with square free conductor.
    Then every solvable Sylow-metacyclic group is admissible over K.



    With the hypothesis on the conductor, we have 
    K ⊆(ζ_m)
    for a square free integer m.
    Therefore K ∩(ζ_p^2)
     = K ∩(ζ_p)
     for every odd prime p,
     and K ∩(ζ_8) =.
     So the hypotheses
     of part (ii) and (iii)
     of Theorem <ref> are satisfied,
     and the result follows.



    Let K = (ζ_m)
    be a cyclotomic field.
    Then every solvable Sylow-metacyclic group is tamely
    admissible over K
    if and only if m
    is square free.


    If m is square free then
    then by the previous 
    corollary
    every solvable Sylow-metacyclic group is tamely
    admissible over (ζ_m).
    On the other hand,
    if p^2 divides m for some
    prime p then 
    (α_p) ⊂(ζ_p^2) ⊆ K = (ζ_m)
    where α_p is as
    in Theorem <ref>.
    By Theorem <ref>, not every solvable
    Sylow-metacyclic group
    is tamely admissible over K.




§ RESULTS OVER GENERAL NUMBER FIELDS


In the previous section,
we largely focused on the case
of adequate extension
being tamely ramified.
Over , this is not 
really a
restriction, at least for 
solvable groups.
This is because Sonn's proof 
(<cit.>) shows that
every -admissible solvable
group is in fact tamely 
admissible. But for a general
number field this may not be true.
We start with the following result
which follows immediately
from Schacher's criterion
(Criterion <ref>):


    Let K be a global field
    and L/K be a G-Galois
    adequate extension which
    is tamely ramified. Then
    G must be Sylow-metacyclic.


    Let p be a prime number
    that divides the order of
    G.
    By Schacher's criterion,
    there is a place v of K
    such that the the Galois
    group of the
    local extension L_v/K_v
    (i.e., the decomposition group) contains a p-Sylow
    subgroup G_p of G.
    
    By taking the fixed field
    of G_p, we get a 
    G_p-Galois
    extension L_v/L_v^G_p.
    By the hypothesis,
    this is a tamely ramified
    extension. But the Galois
    group of a tamely ramified
    finite field extension
    of a local field is 
    metacyclic (see Lemma <ref>).
    Therefore H is metacyclic.

    Since this is true for 
    every prime
    p that divides the order of 
    the group G, it follows that G is a Sylow-metacyclic group.




In this section, we allow our 
adequate extensions to be wildly ramified, in order
to go beyond Sylow-metacyclic
groups.
Admissible groups are often characterized
in terms of generators of their p-Sylow subgroups.
To that end, if G is a p-group,
let d(G) denote the minimum
number of generators of G.
Since G is a p-group,
every minimal
generating
set of G
has d(G)
elements by 
the Burnside basis
theorem. 
Here, “minimal”
is with respect to
the partial ordering
on sets given by
inclusion.




    For a number field K/, 
    let Σ_K denote the set of
    places (inequivalent valuations) of K.
    If the extension K/ is a Galois field extension
    then let e_p denote the ramification degree
    and f_p denote the
    residue degree of 
    a prime p.


 
    Let K be a number field.
    Let p be an odd rational prime
    that decomposes in K and 
    such that p = _1^e_1_2^e_2…_m^e_m
    in K with
    
    [K__1:_p] ≥ [K__2:_p] ≥…≥ [K__m:_p].

    If ζ_p ∉ K__i for i = 1, …, m
    then a p-group G is
    K-admissible 
    if and only if 
    d(G) ≤ [K__2:_p] + 1.




    Suppose that G is a K-admissible p-group,
    and L/K is a K-adequate G-Galois
    extension.
    If G = {1} then the conclusion is true,
    so assume that |G| > 2 since p is an odd prime.
    By Schacher's criterion,
    there are at least two places
    of K such that the decomposition
    group at these places is the whole group G.
    Since |G| > 2, these places are necessarily
    non-archimedean. Let k_1, k_2 be the completion
    of K at any two such places, and
    let l_1/k_1 and l_2/k_2 be the local Galois extensions
    coming from the global extension
    L/K.  Note that the valuation corresponding to k_1 (and k_2)
    might have more than one prolongation to L, but the completion
    of L over each of those prolongations will be isomorphic
    to l_1/k_1 (and l_2/k_2, respectively) since
    L/K is a Galois extension.
    
    If one of these local fields, say k_1,
    has residue characteristic different from p
    then the extension l_1/k_1
    is tamely ramified, and therefore
    G is a metacyclic group. In particular,
    d(G) ≤ 2, and so the conclusion is true.

    If both k_1 and k_2 have residue characteristic
    equal to p, then they are one of the fields 
    K__i for i = 1, …, m.
    Without loss of generality assume
    that [k_1:_p] ≥ [k_2:_p].
    Since ζ_p ∉ k_2,
    By a result of Shafarevich (Theorem 3 in II.5, <cit.>),
    the absolute Galois group of the maximal p-extension
    of k_2 is a free prop-p group on [k_2:_p] + 1
    generators. Since G is a quotient of such a free pro-p
    group, d(G) ≤ [k_2:_p] + 1 ≤ [K__2:_p] + 1.
    This proves one direction of the theorem.

    For the other direction, let
    G be a p-group with
    d(G) ≤ [K__2:_p] + 1.
    Let k_1 = K__1
    and k_2 = K__2.
    Once again, by Theorem 3 in
    II.5 of <cit.>,
    there exist G-Galois
    extensions l_1/k_1, l_2/k_2
    over the local fields k_1, k_2.
    Since the local field k_1 does
    not have a primitive p-th root of
    unity, neither does the global field K.
    Therefore the hypothesis of Neukirch's generalization
    of Grunwald-Wang theorem (Corollary 2 in <cit.>)
    are satisfied, and
    so there exists a G-Galois global extension L/K
    that has l_1/k_1, l_2/k_2 as completions.
    By Schacher's criterion, L/K is a K-adequate extension,
    and therefore G is K-admissible.









 
    Let K be a number field.
    Let p be an odd rational prime
    that decomposes in K and 
    such that p = _1^e_1_2^e_2…_m^e_m
    in K.
    The local fields K__i for i = 1, …, m
    do not contain a primitive p-th root
    of unity in each of the following situations,
    and therefore the conclusion
    of Theorem <ref> is valid:
    
        
  * The prime p is unramified in K/;
        
  * (p-1) ∤ [K__i:_p]
        for i = 1, …, m;
        
  * K/ is Galois, and (p-1) ∤ [K:].
    


    Since p is odd,
    the extension _p(ζ_p)/_p
    of local fields
    is totally ramified.
    Therefore, if the prime p is unramified in K/
    then ζ_p ∉ K__i. This proves part (i).
    The fact that the extension 
    _p(ζ_p)/_p has
    degree p-1 
    shows part (ii) and part (iii) as well.



    It follows that for 
    a number
    field K, 
    away from a set of finitely
    many primes (for example, the set of ramified primes),
    K-admissible p-groups
    are completely determined
    by Theorem <ref>.
    Moreover, a group G
    for which each prime p dividing
    the order of G satisfies the
    conditions 
    in Theorem <ref>
    is K-admissible.
    For example, in case of Galois
    number fields, we get the following 
    Proposition <ref>.



    For a finite group G, let G_p denote
a p-Sylow subgroup of G
(all such subgroups are conjugates 
of each other and hence isomorphic).
For a number field K that is Galois over ,
let K_p denote the
completion of K at a valuation extending the
p-adic valuation (all such completions are isomorphic over _p
since K is assumed to be Galois over ).




 
    Let K be a Galois number field.
    Let G be an odd order group such that
    for each p dividing |G|, 
    
        
  * p decomposes in K.
        
  * Either p is unramified in K,
        or (p-1) ∤ [K:].
        
  * d(G_p) ≤ [K_p : _p]+1
    
    Then G is K-admissible.



    Let G be such a group.
    Since G has odd order, it is a solvable group.
    Let p be a rational prime such that p | |G|.
    Since p decomposes in K, there are
    at least two inequivalent prolongation (extension)
    of the p-adic valuation to K.
    Let k_1, k_2 be the completition of K
    at any of these two inequivalent extensions.
    
    If p is unramified in K or
    (p-1) ∤ [K:] then
    as in Theorem <ref>,
    k_1,k_2 do not contain a primitive p-th
    root of unity. 
    Therefore, by Theorem 3 in II.5, <cit.>,
    the absolute Galois group of the maximal p-extension
    of k_i, i = 1,2 is a free pro-p group on [K_p:_p] + 1
    generators.
    Since d(G_p) ≤ [K_p:_p] + 1,
    there exist G_p-Galois extensions of local fields
    l_1/k_1 and l_2/k_2. Also note that
    ζ_p ∉ since ζ_p ∉ k_1.

    Similarly, for each p | |G|, we can get these G_p-Galois
    local extension over two distinct completions
    of K.  Since G_p ↪ G
    and ζ_p ∉ K, the hypotheses of 
    Corollary 3 of <cit.> are satisfied.
    Thus there is a G-Galois
    global extension L/K 
    that realizes
    these G_p-Galois
    extensions of local
    fields at the completions.
    Therefore, Schacher's criterion implies
    that L/K is a K-adequate
    extension,
    and thus G is K-admissible.


The above theorem provides 
sufficient conditions for a group
to be K-admissible.
Unlike the case of rational numbers,
the question of necessary conditions
remains open for general number fields K,
once we go beyond p-groups and allow
wildly ramified adequate extensions.
But in some special cases the above conditions
are also necessary.
For example,
in the case of nilpotent groups
we can say more due to the following lemma
which follows from taking the tensor products of
appropriate division algebras:



    A nilpotent group G is admissible over a global field
    if and only if all of its Sylow subgroups are.


This leads to the following result.

    Let K be a finite Galois extension of ,
    and G be an odd nilpotent group with
    |G| coprime to the discriminant of K.
    Then G is admissible over K if and only if
    for each p | |G| one of the following two
    conditions holds:
    
    
  * prime p decomposes in K
    and
    d(G_p) ≤ f_p + 1, or,
    
  * prime p does not decompose in K
    and G_p is metacyclic.
    



    For a general nilpotent group,
    Lemma <ref> reduces
    it to the case of p-groups.
    So assume that G is a p-group
    for some odd prime number p.
    The prime p is unramified in K
    by the hypothesis that |G| is coprime
    to the discriminant of K.
    If p decomposes in K
    then by Theorem <ref>, 
    G is K-admissible if
    and only if d(G) ≤ f_p + 1.
    If p does not decompose in K,
    then by Theorem 10.2 of <cit.>,
    G is metacyclic. Conversely,
    by Theorem <ref>,
    a metacyclic p-group is admissible over K.
    This proves the corollary for an odd p-group G.




For a given number field K,
the above results potentially leave out
a finite set of primes for 
the admissibility
of p-groups. 
If such a prime p does not decompose in
K then any admissible
p-group is necessarily
metacyclic.
In that case, one could
use Theorems <ref>
and <ref>
to check when a metacyclic
p-group is 
admissible over K.


On the other hand, if such a prime p decomposes 
in K then an admissible
group need not be metacyclic,
and that situation
is not completely understood.
Nevertheless, we can obtain partial results.
Theorem 10.1 of <cit.> shows that if a p-group G is
admissible over a Galois number field K with [K:] = n, 
then d(G) ≤ (n/2) + 2. 
The following result can be seen as a strengthening of 
this theorem.




    Let K be a finite Galois extension of , and p
    be an odd rational prime such that ζ_p ∉ K,
    and p decomposes in K. Let G be a p-group. Then
    
        
  * If ζ_p ∉ K_p
            then G is K-admissible
            if and only if d(G) ≤ [K_:_p] + 1.
        
  * If ζ_p ∈ K_p
            then G is K-admissible
            if and only if G 
            can be generated by [K_:_p] + 2
            many generators x_1, x_2, …, x_n
            satisfying the relation
            
    x_1^p^s[x_1,x_2][x_3,x_4] … [x_n-1,x_n] = 1

            where p^s is such that ζ_p^s∈ K_p 
            but ζ_p^s+1∉ K_p.
    



    The case when ζ_p ∉ K_p
    follows from Theorem <ref>.
    So assume that ζ_p ∈ K_p,
    and let p^s be the largest power of p
    such that ζ_p^s∈ K_p.
    Since [(ζ_p): _p] = p-1,
    we get that n = [K_p:_p] + 2
    is an even number,
    and n ≥ 4.

    Let G be a K-admissible p-group.
    If G is metacyclic and generated by g_1
    and g_2,
    then the free pro-p 
    group F_n on n generators
    x_1, … x_n has a 
    surjective map to G by
    x_2 ↦ g_1, x_4 ↦ g_2
    and x_i ↦ 1, i ≠ 2,4.
    Clearly, this map satisfies the relation 
    x_1^p^s[x_1,x_2][x_3,x_4] … [x_n-1,x_n] = 1.

    Now consider the case when G is not metacyclic.
    Let L/K be a G-Galois K-adequate extension.
    By Schacher's criterion, there will be two distinct
    places of K for which the decomposition group
    corresponding to the adequate extension L/K
    will be the whole group G. Let k
    be the completion of K at one such place,
    and l/k be a corresponding G-Galois extension 
    of local fields coming from the extension L/K
    (since L/K is Galois, all such local extensions
    over k will be k-isomorphic).
    Since tamely ramified Galois extensions of local
    fields have metacyclic 
    Galois groups and G is not metacyclic,
    the residue characteristic of k
    must be p, i.e. k ≅ K_p.
    Since ζ_p ∈ k by assumption,
    the absolute Galois group of the maximal p-extension
    of k is a Demuškin pro-p
     of rank [k:_p]+2 (Theorem 4 in <cit.>).
    In particular, it's the  pro-p group
    on [k:_p]+2 generators x_1, …, x_n
    with one relation 
    x_1^p^s[x_1,x_2][x_3,x_4] … [x_n-1,x_n] = 1
    (Theorem 7 of <cit.>),
    and the result follows.

    In the other direction,
    assume that G is finite p-group
    that can be generated by [k:_p]+2
    many generators subject to the given relation.
    Since p decomposes in K, there are at least
    two distinct completions k_1, k_2
    with residue characteristic p.
    Once again by Theorem 7 of
    <cit.>, there exist G-Galois
    local extensions l_1/k_1 and l_2/k_2.
    Since ζ_p ∉ K,
    Corollary 3 of <cit.> asserts the existence
    of a G-Galois global field extension L/K
    that has l_i/k_i, i = 1,2
    as completions. 
    By Schacher's criterion this suffices 
    to show that L/K is K-adequate
    and G is admissible over K.


Note that since we assumed p
to be an odd prime, if ζ_p ∈ K_p
then n = [K_p:_p] is divisible by (p-1).
In particular, it is an even number
and the above description makes sense.

Adapting the proof of Theorem <ref>
we get a result in the converse direction
of Theorem 10.1 of <cit.>:

    Let K be a finite Galois extension of , and 
    G be an odd order group such that for each 
    prime p that divides |G|, 
    the prime p decomposes in the number
    field K and K does not have a primitive p-th root of unity.
    If d(G_p) ≤ ([K_p:_p]/2) + 1
    for each Sylow p-subgroup G_p
    then G is K-admissible.


    By Schacher's criterion,
    it suffices to construct a G-Galois
    field extension L/K such that
    for each prime p dividing |G|,
    there are two places of K for which the 
    decomposition group is a p-Sylow
    subgroup of G. Since ζ_p ∉ K
    for each prime p dividing |G|,
    the hypothesis of Corollary 3 in <cit.>
    are satisfied. Therefore it suffices
    to show that for each prime p
    dividing the order of G, 
    there are two (distinct) completions
    of K that admit G_p-Galois
    field extensions, where G_p is a
    p-Sylow subgroup of G.

    So let p divide |G|, and since
    p decomposes in K, let k_1, k_2
    be two distinct completions of K
    with respect to valuations extending 
    the p-adic valuation.
    By hypothesis, d(G_p) ≤ ([k_i:_p]/2)+1
    for each i = 1,2.
    If ζ_p ∉ k_1 (and so ζ_p ∉ k_2
    either since k_1 ≅ k_2 over _p)
    then by Theorem 3, the absolute Galois
    group of the maximal p-extension 
    of k_i, i = 1,2 is a free pro-p
    group on [K_p:_p]+1 generators.
    In particular, k_1,k_2
    admit G_p-Galois field extensions.

    On the other hand, if ζ_p ∈ k_1 
    (and so also in k_2)
    then the absolute Galois group of the maximal
    p-extension of k_i has a presentation
    with [K_p:_p] + 2 generators
    x_1, …, x_n with one relation
    x_1^p^s[x_1,x_2][x_3,x_4] … [x_n-1,x_n] = 1
    (Theorems 7.5.11
    and 7.5.12,
    <cit.>).
    Sending each x_i for i odd number
    to 1 gives
    a surjection to a free pro-p
    group on ([K_p:_p]/2) + 1
    generators, and thus there is a
    G_p-Galois extension of local fields over
    both k_1 and k_2.
    This finishes the proof.



The proof of the above theorem uses a result in
<cit.> that generalizes the
Grunwald-Wang theorem, and it also uses the description of 
the Galois
group of maximal p-extension of
local fields as Demuškin groups <cit.>,
i.e., Poincaré groups of dimension 2.
The presentations of these groups have a striking
similarity to that of pro-p completion of fundamental groups
of topological surfaces,
and one might ask
whether that analogy 
in the sense of arithmetic topology 
can be useful
in providing an alternative description
of admissible groups in this case.

Similar to the case of Proposition <ref>, this result partially 
extends to more general solvable groups, as well as to non-Galois number fields.





    In the case that G is admissible
    and p does not decompose in K (i.e.,
    the p-adic valuation on  extends uniquely
    to K),
    one of the two places 
    in Schacher's criterion must have residue characteristic different from
    p. This forces G to be metacyclic, and the characterization of
    admissible groups in 
    that case is already known (see <cit.>).






§ ADMISSIBILITY OF TEXT-GROUPS OVER SPECIAL CLASSES OF NUMBER FIELDS


This section contains results about
admissibility of p-groups 
after specializing to certain classes of
number fields, such as Galois number fields, 
number fields of degree 2^n and odd degree over ,
and finally the cyclotomic fields.


As a corollary to Theorem <ref> and
Theorem <ref>
we get the following result
for number fields that are Galois over .
Here f_p is the residue degree of prime p.

 
	Let K be a Galois number field.
	An odd p-group with 
	p coprime to  the discriminant
	of K/ is K-admissible if and only if
	one of the following conditions holds:
	
		
  * prime p decomposes in K and d(G) ≤ f_p+1, or,
		
  * prime p does not decompose in K and G is metacyclic.
	


A special class of Galois number fields are
the cyclotomic number fields of 
type K = (ζ_l^r) for l a prime number.
Since l is the only ramified prime
in K/, Corollary <ref> leaves
out only the case of l-groups. 
Since l does not decompose in K, any admissible
l-group must be metacyclic by Theorem 10.2 of <cit.>.
As far as the K-admissibility of l-metacyclic
group is concerned, 
it depends on the field.
Every metacyclic l-group
is known to be admissible over (ζ_l),
for example, by
the discussion following Proposition 32
in <cit.>, or by Corollary <ref>.
But it follows from Prop <ref>
that there are metacyclic l-groups
that are not admissible over (ζ_l^r)
for r ≥ 2.  




 §.§ Number fields of degree 2^n

Specializing further to number fields
Galois over  that have degree [K:]
a power of 2, we have

 
	Let K be a Galois
	number field of degree 2^n,
	and G be an odd p-group.
	Then the following assertions hold:
	
		
  * If p does not decompose in K, 
		then G is K-admissible
		if and only if G is metacyclic.
		
  * If p decomposes in K, 
		and either (p-1) ∤ [K_p:_p]
		or p is unramified in K
		then G is K-admissible
		if and only if d(G) ≤ [K_p:_p] + 1.
	



    Consider first the case when p does not
    decompose in K.    
    The prime p does not divide [K:]
    since p is odd,
    and so the result follows from Corollary <ref>.

    The case when p decomposes in K
    follows from Proposition <ref>.




    Note that in order for (p-1)
    to divide the local degree [K_p:_p]
    which is a power of 2,
    p must be a Fermat prime and smaller
    than or equal to [K_p:_p]/2.
    At the time of writing this manuscript,
    only 5 Fermat primes are
    known (namely, 3, 5, 17, 257, 65537)
    and this list is conjectured
    to be exhaustive.


Since every quadratic extension is automatically
Galois, we can use the above corollary in that case.
Moreover, there are no exceptional Fermat primes
for quadratic extensions,
and thus we get a complete
characterization of admissible p-groups for odd primes p.

 
	Let K be a quadratic number field,
	and G be an odd p-group for some rational prime p.
	Then G is K-admissible
	if and only if one of the following conditions
	holds:
	
		
  * prime p decomposes in K and d(G) ≤ 2, or,
		
  * prime p does not decompose in K and G is metacyclic.
	



    Apply Corollary <ref>
    and observe that if a prime p splits in K
    then [K_p:_p] = 1.



    The above corollary leaves out the case of 
    2-groups. We point out that there
    are examples of quadratic number field K
    and metacyclic 2-groups that are
    not admissible over K.
    For example, the dihedral group
    of order 8 is known to 
    not be (i)-admissible.
    (It follows from Corollary <ref>, for example)


The next group of number fields 
with degree a power of two are quartic number fields,
and that case is more involved than the case of quadratic 
number fields.
First, the field can be non-Galois,
and second, there is the possible Fermat prime 3
even if the field is Galois over .
When the field is non-Galois, 
our strategy is to look at the various possible splittings of 
primes, and argue that the local
field cannot contain p-th roots of 
unity. The precise result is

 
	Let K be a quartic number field.
	Then a p-group G for 
	p ≠ 2,3 is admissible over K
	if and only if one of the following two conditions hold:
	
		
  * p does not decompose in K, and G is metacyclic.
		
  * p decomposes in K, and d(G) ≤| pmin([K_:_p]) + 1.
	



    Note that the case when K/ is
    Galois follows from Corollary <ref>
    after observing the following two points.
    First, that the only exceptional Fermat prime
    in this case is 3, which is excluded from the
    statement.
    Second, if the prime p decomposes in K
    then  [K_:_p] is same for each
     extending the p-adic valuation.

    The general case is proven with a similar argument
    as in the Corollary <ref>.
    We start with the case
    when p does not 
     decompose in K.
     In that case, since p
     does not divide 4 = [K:], the result
    follows from Corollary <ref>.
    
    In the case that p decomposes in K, 
    we argue that none of the completions
    at p contain ζ_p. 
    Let | p, and there are the following two
    subcases.
     
        
  * K_/_p is unramified.
        Since p is odd, and for odd primes 
        _p(ζ_p)/_p is ramified.
        It follows that K_ does not
        contain a primitive p-th root of unity.
        
        
  * If K_/_p
        is ramified then the
        ramification degree can only be 2
        or 3 since we assumed that p decomposes
        in K and [K:] = 4.
        If ζ_p ∈ K_
        then the ramification degree must
        be at least four since p ≥ 5
        and [_p(ζ_p) : _p] = p-1.
        Therefore ζ_p ∉ K_.
    

    So the hypothesis of Theorem <ref>
    is satisfied, and | pmin([K_:_p])
    equals the second largest degree of local extensions
    as in Theorem <ref>.





 §.§ Odd degree number fields

Similar to Corollary <ref>,
the Galois number fields of odd degree
are another special class of number fields where
we can prove stronger results.


 
    Let K be a Galois number field whose degree [K:] is an odd number,
    and G be an odd p-group.
    Then the following assertions hold:
    
        
  * If p does
        not decompose in K
        then G is K-admissible
        if and only if G has
        a Liedahl presentation for K. (See <ref>
        for a definition of
        Liedahl presentation.)
        
            
  * Moreover,
            if in addition p
        is tamely ramified in K, then
        G is K-admissible if and only
        if it is metacyclic.
        
        
  * If p decomposes in K, 
        then G is K-admissible
        if and only if
        d(G) ≤ [K_p : _p] + 1.
    



    The case when p does not decompose in K
    follows from Liedahl's result (Theorem <ref>) and Corollary <ref>.

    By the hypothesis, the
    degree of the number
    field [K:]
    is odd, whereas (p-1) is
    an even integer since 
    p is assumed
    to be odd.
    It follows that 
    (p-1) ∤ [K:].
    The case where p 
    decomposes in K 
    now follows
    from Theorem <ref> in light of the 
    observation that
    K/ is a Galois 
    extension 
    and (p-1) ∤ [K:].


The first odd degree case is the case of cubic
number fields. Arguing similar to the case
of quartic number fields, we get


	
    Let K be a cubic number field,
    and G be a p-group for p ≠ 2,3. 
    Then G is K-admissible
    if and only if one of the following conditions
    holds:
    
        
  * prime p does not decompose in K and G is metacyclic.
        
  * prime p decomposes in K and d(G) ≤ 2.
    



    Since
    p ≠ 2,3,
    the case when K/ is Galois follows from
    Theorem <ref> once we
    observe that if p decomposes
    in K then K_p = _p.

    Next consider the case when K/
    is not Galois.
    If p does not decompose in K
    then the result once again follows from
    Corollary <ref>.
    Finally, consider the case that p decomposes
    in K, and let k be any completion of K
    for a valuation extending the p-adic
    valuation on .
    We have [_p(ζ_p):_p] = p-1 ≥ 4 
    since p ≥ 5, and so
    ζ_p ∉ k since [k:_p] ≤ 3.
    Therefore, we can invoke Theorem <ref>.
    The result follows once we observe that for a cubic
    number field the second biggest local degree is necessarily
    1 in Theorem <ref>.    



The exceptional case of  p = 3 in
Proposition <ref>,
and more generally 
the case of p | [K:] in 
Theorem <ref>, can have 
a more involved description of admissible p-groups.
An example of this phenomenon is 
Lemma <ref>,
where it was shown that the non-abelian
    semi-direct product /l^2 ⋊/l
    is not admissible over the unique
    degree l number field K
    inside (ζ_l^2).
In particular, /9 ⋊/3 is not admissible over 
the cubic number field (ζ_9+ζ_9^-1).














Author Information:

Deependra Singh

Department of Mathematics, Emory University, Atlanta, GA 30322, USA

email: deependra.singh@emory.edu



