### Genus group

In algebraic number theory, the genus field G of a 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 pi run over the prime factors of D. For each such prime p, define p* as follows:

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

Then the genus field is the composite K(√pi*).