en>Tony1 at 12:28, 26 October 2011

2011-10-26T12:28:17Z

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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''.

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: this definition was used by Leopoldt and Hasse.

If ''K''='''Q'''(√''m'') (''m'' squarefree) is a quadratic field of discriminant ''D'', the genus field of ''K'' is a composite of quadratic fields. Let ''p''<sub>''i''</sub> run over the prime factors of ''D''. For each such prime ''p'', define ''p''* as follows:

:<math> p^* = \pm p \equiv 1 \pmod 4 \text{ if } p \text{ is odd} ; </math>
:<math> 2^* = -4, 8, -8 \text{ according as } m \equiv 3 \pmod 4, 2 \pmod 8, -2 \pmod 8 . </math>

Then the genus field is the composite ''K''(√''p''<sub>i</sub>*).

==See also==
* [[Hilbert class field]]

==References==
* {{cite book | zbl=0353.12001 | last=Ishida | first=Makoto | title=The genus fields of algebraic number fields | series=Lecture Notes in Mathematics | volume=555 | publisher=[[Springer-Verlag]] | year=1976 | isbn=3-540-08000-7 }}

[[Category:Class field theory]]

{{numtheory-stub}}

Constant-energy surface - Revision history

en>Tony1 at 12:28, 26 October 2011