Abelian extension

The development of class field theory has provided detailed information about abelian extensions of number fields[?], function fields of algebraic curves over finite fields, and local fields.

In general extensions formed by adjoining any roots of unity are abelian. If a field K already contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting so-called Kummer extension[?] is an abelian extension (if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension[?]). In general, however, the Galois groups of n-th roots of elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-direct product. The Kummer theory[?] gives a complete description of the abelian extension case.

There is an important analogy with the fundamental group in topology, which classifies all covering spaces of a space: abelian covers are classified by its abelianisation[?] which relates directly to the first homology group.