Genus 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(√m) (m squarefree) is a quadratic field of discriminant D, the genus field of K is a composite of quadratic fields. Let pi run over the prime factors of D. For each such prime p, define p∗ as follows:
- [math]\displaystyle{ p^* = \pm p \equiv 1 \pmod 4 \text{ if } p \text{ is odd} ; }[/math]
- [math]\displaystyle{ 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]\displaystyle{ K(\sqrt{p_i^*}). }[/math]
See also
References
- Ishida, Makoto (1976). The genus fields of algebraic number fields. Lecture Notes in Mathematics. 555. Springer-Verlag. ISBN 3-540-08000-7.
- Janusz, Gerald (1973). Algebraic Number Fields. Pure and Applied Mathematics. 55. Academic Press. ISBN 0-12-380250-4.
- Lemmermeyer, Franz (2000). Reciprocity laws. From Euler to Eisenstein. Springer Monographs in Mathematics. Berlin: Springer-Verlag. ISBN 3-540-66957-4. https://books.google.com/books?id=EwjpPeK6GpEC.
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