Difference between revisions of "Abelian number field"
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| − | 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 $\mathbf{Q}(\zeta_n)$, $\zeta^n=1$. | + | 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]]. | 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]]. | ||
Revision as of 16:18, 18 September 2017
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.
References
- 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: http://encyclopediaofmath.org/index.php?title=Abelian_number_field&oldid=39903
Abelian number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abelian_number_field&oldid=39903
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article