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:

$p^{*}=\pm p\equiv 1(\mathrm{mod}4)\text{ if }p\text{ is odd};$

$2^{*}=-4,8,-8\text{ according as }m\equiv 3(\mathrm{mod}4),2(\mathrm{mod}8),-2(\mathrm{mod}8).$

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

See also

• Hilbert class field