Genus field


Genus field

In algebraic number theory, the genus field "G" of a 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 ["G":"K"] 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.

ee also

* Hilbert class field

References

*


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