title: Genus field
text: # 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:
  
      $p^{*}=\pm p\equiv 1{\pmod {4}}{\text{ if }}p{\text{ is odd}};$
      $2^{*}=-4,8,-8{\text{ according as }}m\equiv 3{\pmod {4}},2{\pmod {8}},-2{\pmod {8}}.$
  
  Then the genus field is the composite $K({\sqrt {p_{i}^{*}}}).$
  
  