Genus field

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.