View source for Abelian number field

An Abelian [[Extension of a field|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 field]]s $\mathbf{Q}(\zeta_n)$, $\zeta^n=1$.

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]].

See [[Class field theory]].  

====References====
*Lawrence C. Washington, " Introduction to Cyclotomic Fields" (2 ed) Graduate Texts in Mathematics '''83''' Springer (2012) ISBN 1461219345