Editing Abelian extension (section)

==Description==
[[Class field theory]] provides detailed information about the abelian extensions of [[number field]]s, [[function field of an algebraic variety|function fields]] of [[algebraic curve]]s over finite fields, and [[local field]]s.

There are two slightly different definitions of the term '''cyclotomic extension.''' It can mean either an extension formed by adjoining [[roots of unity]] to a field, or a subextension of such an extension. The [[cyclotomic field]]s are examples. A cyclotomic extension, under either definition, is always abelian.

If a field ''K'' contains a primitive ''n''-th root of unity and the ''n''-th root of an element of ''K'' is adjoined, the resulting [[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, and the [[Kronecker–Weber theorem]] tells us that if ''K'' is the field of [[rational number]]s, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.

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]].

{{Further|Ring class field}}