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($\sqrt{m}$) (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&lowast; 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 $$K(\sqrt{p_i^*}).$$