abelian number field

Definition 1.

An abelian number field is a number field $K$ such that $K/\mathbb{Q}$ is an abelian extension, i.e. $K/\mathbb{Q}$ is Galois and $\operatorname{Gal}(K/\mathbb{Q})$ is an abelian group.

The abelian number fields are classified by the Kronecker-Weber Theorem.

Definition 2.

A cyclic number field is an (abelian) number field $K$ such that $K/\mathbb{Q}$ is a Galois extension and $\operatorname{Gal}(K/\mathbb{Q})$ is a finite cyclic group (therefore abelian).

Title abelian number field AbelianNumberField 2013-03-22 16:01:24 2013-03-22 16:01:24 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 11-00 GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ cyclic number field