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^∗).

See also

• Hilbert class field

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.