Abelian number field

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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.

See Class field theory.


  • Lawrence C. Washington, " Introduction to Cyclotomic Fields" (2 ed) Graduate Texts in Mathematics 83 Springer (2012) ISBN 1461219345
How to Cite This Entry:
Abelian number field. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article