Genus field

In algebraic number theory, the genus field G 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 [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_i run over the prime factors of D. For each such prime p, define p^∗ as follows:

p^{*}=\pm p\equiv 1{\pmod 4}{\text{ if }}p{\text{ is odd}};

2^{*}=-4,8,-8{\text{ according as }}m\equiv 3{\pmod 4},2{\pmod 8},-2{\pmod 8}.

Then the genus field is the composite K(√p_i^∗).

References

Ishida, Makoto (1976). The genus fields of algebraic number fields. Lecture Notes in Mathematics. 555. Springer-Verlag. ISBN 3-540-08000-7. Zbl 0353.12001.
Janusz, Gerald (1973). Algebraic Number Fields. Pure and Applied Mathematics. 55. Academic Press. ISBN 0-12-380250-4. Zbl 0307.12001.
Lemmermeyer, Franz (2000). Reciprocity laws. From Euler to Eisenstein. Springer Monographs in Mathematics. Berlin: Springer-Verlag. ISBN 3-540-66957-4. MR 1761696. Zbl 0949.11002.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Genus field

See also

References