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 p_i run over the prime factors of D. For each such prime p, define p^∗ as follows:

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

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

Then the genus field is the composite ${\displaystyle K({\sqrt {p_{i}^{*}}}).}$

• Hilbert class field

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• Janusz, Gerald (1973). Algebraic Number Fields. Pure and Applied Mathematics. Vol. 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.