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In algebraic number theory, the genus field $\Gamma (K)$ 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 $[\Gamma (K):K]$ and the genus group is the Galois group of $\Gamma (K)$ 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 $K=\mathbb {Q} ({\sqrt {m}})$ ($m$ square-free) 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 $K({\sqrt {p_{i}^{*}}}).$

See also

• Hilbert class field

References

• Ishida, Makoto (1976). The Genus Fields of Algebraic Number Fields. Lecture Notes in Mathematics. Vol. 555. Berlin, Heidelberg: Springer. ISBN 978-3-540-08000-8. Zbl 0353.12001.

• Janusz, Gerald J. (1973). Algebraic Number Fields. Pure and Applied Mathematics. Vol. 55. New York: Academic Press. ISBN 0-12-380250-4. Zbl 0307.12001.

• Lemmermeyer, Franz (2000). Reciprocity Laws: From Euler to Eisenstein. Springer Monographs in Mathematics. Berlin, Heidelberg: Springer. ISBN 978-3-540-66957-9. MR 1761696. Zbl 0949.11002.