Genus field

IfK=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:

${\displaystyle p^{*}=\pm p\equiv 1{\pmod {4}}{\text{ if }}p{\text{ is odd}};}$

${\displaystyle 2^{*}=-4,8,-8{\text{ according as }}m\equiv 3{\pmod {4}},2{\pmod {8}},-2{\pmod {8}}.}$

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

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