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[parent] abelian number field (Definition)
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).



"abelian number field" is owned by alozano.
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See Also: Galois groups of finite abelian extensions of $\mathbb{Q}$

Also defines:  cyclic number field

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Cross-references: cyclic group, finite, Galois extension, abelian, Kronecker-Weber theorem, abelian group, abelian extension, number field
There are 5 references to this entry.

This is version 2 of abelian number field, born on 2006-06-20, modified 2006-07-19.
Object id is 8063, canonical name is AbelianNumberField.
Accessed 1040 times total.

Classification:
AMS MSC: 11-00 (Number theory :: General reference works )
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