Genus field

Template:Short description In algebraic number theory, the genus field $\mathrm{\Gamma }(K)$ 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 $[\mathrm{\Gamma }(K):K]$ and the genus group is the Galois group of $\mathrm{\Gamma }(K)$ 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 $K=\mathbb{Q}(\sqrt{m})$ ($m$ square-free) 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:

$p^{\ast }=\pm p\equiv 1(\mathrm{mod}4)\text{ if }p\text{ is odd};$

$2^{\ast }=-4,8,-8\text{ according as }m\equiv 3(\mathrm{mod}4),2(\mathrm{mod}8),-2(\mathrm{mod}8).$

Then the genus field is the composite $K(\sqrt{p_i^{\ast }}).$

See also

• Hilbert class field

References

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