Genus field

In algebraic number theory, the genus field G 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 and the genus group is the Galois group of G over K. In algebraic number theory, the genus field G 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 and the genus group is the Galois group of G 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 K ( p i ∗ ) . {displaystyle K({sqrt {p_{i}^{*}}}).}