{{Short description|Maximal abelian extension of an algebraic number field}} In algebraic number theory, the '''genus field''' ''Γ(K)'' of an algebraic 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 [''Γ(K)'':''K''] and the '''genus group''' is the Galois group of ''Γ(K)'' 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'''({{radic|''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''<sup>∗</sup> 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 <math>K(\sqrt{p_i^*}).</math>
==See also== * Hilbert class field
==References== * {{cite book | 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 | zbl=0353.12001 }} * {{cite book | first=Gerald | last=Janusz | title=Algebraic Number Fields | year=1973 | publisher=Academic Press | isbn=0-12-380250-4 | series=Pure and Applied Mathematics | volume=55 | zbl=0307.12001 }} * {{cite book | last=Lemmermeyer | first= Franz | title=Reciprocity laws. From Euler to Eisenstein | series= Springer Monographs in Mathematics | publisher=Springer-Verlag | location=Berlin | year= 2000 | isbn= 3-540-66957-4 | url=https://books.google.com/books?id=EwjpPeK6GpEC | mr=1761696 | zbl=0949.11002 }}
Category:Class field theory
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