|
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 ''p''''i'' 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∗). ==See also== * Hilbert class field 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Genus field」の詳細全文を読む スポンサード リンク
|