section_path: Main subdivisions/Algebraic number theory
section_name: Algebraic number theory
paragraph_text: Algebraic number theory
  
  An algebraic number is any complex number that is a solution to some polynomial equation f (x) = 0 {\displaystyle f(x)=0} with rational coefficients; for example, every solution x {\displaystyle x} of x 5 + (11 / 2) x 3 − 7 x 2 + 9 = 0 {\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0} (say) is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields. Algebraic number theory studies algebraic number fields. Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.
  
  It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in Disquisitiones arithmeticae can be restated in terms of ideals and norms in quadratic fields. (A quadratic field consists of allnumbers of the form a + b d {\displaystyle a+b{\sqrt {d}}}, where a {\displaystyle a} and b {\displaystyle b} are rational numbers and d {\displaystyle d} is a fixed rational number whose square root is not rational.)For that matter, the 11th-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.
  
  The grounds of the subject as we know it were set in the late nineteenth century, when ideal numbers, the theory of ideals and valuation theory were developed; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationalsand − 5 {\displaystyle {\sqrt {-5}}}, the number 6 {\displaystyle 6} can be factorised both as 6 = 2 ⋅ 3 {\displaystyle 6=2\cdot 3} and 6 = (1 + − 5) (1 − − 5) {\displaystyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})}; all of 2 {\displaystyle 2}, 3 {\displaystyle 3}, 1 + − 5 {\displaystyle 1+{\sqrt {-5}}} and 1 − − 5 {\displaystyle 1-{\sqrt {-5}}} are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by Kummer) seems to have come from the study of higher reciprocity laws, that is, generalisations of quadratic reciprocity.
  
  Number fields are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K.(For example, the complex numbers C are an extension of the reals R, and the reals R are an extension of the rationals Q.)Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions L of K such that the Galois group Gal(L / K) of L over K is an abelian group —are relatively well understood.Their classification was the object of the programme of class field theory, which was initiated in the late 19th century (partly by Kronecker and Eisenstein) and carried out largely in 1900–1950.
  
  An example of an active area of research in algebraic number theory is Iwasawa theory. The Langlands program, one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.