Genus field

In algebraic number theory, the genus field Γ ( K ) {\displaystyle \Gamma (K)} of an algebraic number field K {\displaystyle K} is the maximal abelian extension of K {\displaystyle K} which is obtained by composing an absolutely abelian field with K {\displaystyle K} and which is unramified at all finite primes of K {\displaystyle K} . The genus number of K {\displaystyle K} is the degree [ Γ ( K ) : K ] {\displaystyle [\Gamma (K):K]} and the genus group is the Galois group of Γ ( K ) {\displaystyle \Gamma (K)} over K {\displaystyle K} . If K {\displaystyle K} is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of K {\displaystyle K} unramified at all finite primes: this definition was used by Leopoldt and Hasse. If K = Q ( m ) {\displaystyle K=\mathbb {Q} ({\sqrt {m}})} ( m {\displaystyle m} square-free) is a quadratic field of discriminant D {\displaystyle D} , the genus field of K {\displaystyle K} is a composite of quadratic fields. Let pi run over the prime factors of D {\displaystyle D} . For each such prime p, define p∗ as follows:

p ∗ = ± p ≡ 1 ( mod 4 ) if p is odd ; {\displaystyle p^{*}=\pm p\equiv 1{\pmod {4}}{\text{ if }}p{\text{ is odd}};}

2 ∗ = − 4 , 8 , − 8 according as m ≡ 3 ( mod 4 ) , 2 ( mod 8 ) , − 2 ( mod 8 ) . {\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 K ( p i ∗ ) . {\displaystyle K({\sqrt {p_{i}^{*}}}).}

Tags