Genus field explained

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 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 if p isodd ; 2 ^{*} = -4 , 8 , -8 accordingas m \equiv 3 \pmod 4 , 2 \pmod 8 , -2 \pmod 8 .

Then the genus field is the composite

K(\sqrt{p ^{*}).} _{i}

See also

• Hilbert class field

References

• Book: Ishida, Makoto . The genus fields of algebraic number fields . Lecture Notes in Mathematics . 555 . . 1976 . 3-540-08000-7 . 0353.12001 .
• Book: Janusz, Gerald . Algebraic Number Fields . 1973 . Academic Press . 0-12-380250-4 . Pure and Applied Mathematics . 55 . 0307.12001 .
• Book: Lemmermeyer, Franz . Reciprocity laws. From Euler to Eisenstein . Springer Monographs in Mathematics . . Berlin . 2000 . 3-540-66957-4 . 1761696 . 0949.11002 .