Genus field

 ( Class Field Theory ) 

In algebraic number theory, the genus field Γ(K) of an algebraic number field K is the Maximal element 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:

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

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

Then the genus field is the composite $K(\backslash sqrt\{p\_i^*\}).$

• Hilbert class field

• Makoto Ishida (1976). 9783540080008, Springer-Verlag. ISBN 9783540080008
• Gerald Janusz (1973). 9780123802507, Academic Press. ISBN 9780123802507
• Franz Lemmermeyer (2025). 9783540669579, Springer-Verlag. . ISBN 9783540669579