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 pi run over the prime factors of D. For each such prime p, define p∗ as follows: Then the genus field is the composite
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