# 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.

#### References

• Lawrence C. Washington, " Introduction to Cyclotomic Fields" (2 ed) Graduate Texts in Mathematics 83 Springer (2012) ISBN 1461219345
How to Cite This Entry:
Kronecker–Weber theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Kronecker%E2%80%93Weber_theorem&oldid=39899