# Abelian number field

An Abelian extension of the field of rational numbers $\mathbf{Q}$, i.e. a Galois extension $K$ of $\mathbf{Q}$ such that the Galois group $\mathrm{Gal}(K/\mathbf{Q})$ is Abelian. Examples include: the quadratic number fields $\mathbf{Q}(\sqrt{d})$ and the cyclotomic fields $\mathbf{Q}(\zeta_n)$, $\zeta^n=1$.
The Kronecker–Weber theorem states that every Abelian number field is a subfield of a cyclotomic field. The conductor of an abelian number field $K$ is the least $n$ such that $K$ is contained in $\mathbf{Q}(\zeta_n)$, cf. Conductor of an Abelian extension.