Genus field: Difference between revisions
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In [[algebraic number theory]], the '''genus field''' ''G'' of a [[number field]] ''K'' is the [[maximal]] [[abelian extension|abelian]] extension of ''K'' which is obtained by composing an absolutely abelian field with ''K'' and which is [[unramified]] at all finite primes of ''K''. The '''genus number''' of ''K'' is the degree [''G'':''K''] and the '''genus group''' is the [[Galois group]] of ''G'' over ''K''. | In [[algebraic number theory]], the '''genus field''' ''G'' of a [[number field]] ''K'' is the [[maximal]] [[abelian extension|abelian]] extension of ''K'' which is obtained by composing an absolutely abelian field with ''K'' and which is [[unramified]] at all finite primes of ''K''. The '''genus number''' of ''K'' is the degree [''G'':''K''] and the '''genus group''' is the [[Galois group]] of ''G'' over ''K''. | ||
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==References== | ==References== | ||
* {{cite book | last=Ishida | first=Makoto | title=The genus fields of algebraic number fields | series=Lecture Notes in Mathematics | publisher=[[Springer Verlag]] | date=1976 | isbn=3-540-08000-7 }} | * {{cite book | last=Ishida | first=Makoto | title=The genus fields of algebraic number fields | series=Lecture Notes in Mathematics | publisher=[[Springer Verlag]] | date=1976 | isbn=3-540-08000-7 }} | ||
Revision as of 15:14, 28 October 2008
In algebraic number theory, the genus field G of a number field K is the maximal abelian extension of K which is obtained by composing an absolutely abelian field with K and which is unramified at all finite primes of K. The genus number of K is the degree [G:K] and the genus group is the Galois group of G over K.
If K is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of K unramified at all finite primes.
See also
References
- Ishida, Makoto (1976). The genus fields of algebraic number fields. Springer Verlag. ISBN 3-540-08000-7.