Abelian extension
In the mathematical subfield of algebra , an Abelian extension is a Galois field extension with an Abelian Galois group . In the special case of a cyclical Galois group, there is a cyclical expansion .
The class field theory describes the Abelian extensions of number fields , function fields of algebraic curves over finite fields and local fields .
Extensions that result from the adjunction of roots of unity are Abelian, for example all algebraic extensions of finite fields . If a body already contains a primitive -th root of unity and the characteristic of is not a divisor of , then every extension by the adjunction of a -th root of an element is of Abelian, called Kummer extension . If one adjoint all -th roots of an element, the extension is generally no longer Abelian, but a semi-direct product , since the Galois group operates on the roots and the -th roots of unity. The Kummer theory describes the Abelian extensions of a body, and the Kronecker-Weber theorem says that the Abelian extensions are exactly those that are contained in the fields of circular division .
