abelian number field

Definition 1.

An abelian number field is a number field $K$ such that $K\mathrm{/}\mathrm{Q}$ is an abelian extension, i.e. $K\mathrm{/}\mathrm{Q}$ is Galois and $\mathrm{Gal}\mathit{}\mathrm{(}K\mathrm{/}\mathrm{Q}\mathrm{)}$ 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 $K\mathrm{/}\mathrm{Q}$ is a Galois extension and $\mathrm{Gal}\mathit{}\mathrm{(}K\mathrm{/}\mathrm{Q}\mathrm{)}$ is a finite cyclic group (therefore abelian).

Title	abelian number field
Canonical name	AbelianNumberField
Date of creation	2013-03-22 16:01:24
Last modified on	2013-03-22 16:01:24
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Definition
Classification	msc 11-00
Related topic	GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ
Defines	cyclic number field