Abelian extension

In the mathematical field of algebra is an abelian extension of a Galois field extension with abelian Galois group. In the special case of a cyclic Galois group there is a cyclic extension.

The class field theory describes the abelian extensions of number fields, function fields of algebraic curves over finite fields and local bodies.

Extensions that emerge by the adjunction of roots of unity, are abelian, so for example, all algebraic extensions of finite fields. When a body already contains a primitive root of unity and the characteristic of not divide is, then every extension by adjoining a - th root of an element of abelian called Kummer extension. Adjoint one all - th roots of an element, so the expansion is in general no longer abelian, but a semidirect product, since the Galois group acts on the roots and the roots of unity. The Kummer theory describes the abelian extensions of a body, and the set of Kronecker -Weber says that for abelian extensions exactly are contained in the cyclotomic bodies.

• Number Theory
• Algebraic Number Theory