Galois group abelian group class field theory Kronecker Eisenstein numbers number fields
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.
