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).