Genus field

In algebraic number theory, the genus field ${\displaystyle \Gamma (K)}$ of an algebraic number field ${\displaystyle K}$ is the maximal abelian extension of ${\displaystyle K}$ which is obtained by composing an absolutely abelian field with ${\displaystyle K}$ and which is unramified at all finite primes of ${\displaystyle K}$. The genus number of ${\displaystyle K}$ is the degree ${\displaystyle [\Gamma (K):K]}$ and the genus group is the Galois group of ${\displaystyle \Gamma (K)}$ over ${\displaystyle K}$.

If ${\displaystyle K}$ is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of ${\displaystyle K}$ unramified at all finite primes: this definition was used by Leopoldt and Hasse.

If ${\displaystyle K=\mathbb {Q} ({\sqrt {m}})}$ (${\displaystyle m}$ square-free) is a quadratic field of discriminant ${\displaystyle D}$, the genus field of ${\displaystyle K}$ is a composite of quadratic fields. Let p_i run over the prime factors of ${\displaystyle D}$. 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 ${\displaystyle K({\sqrt {p_{i}^{*}}}).}$