algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

, the genus field ''Γ(K)'' of an

algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

''K'' is the maximal

abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...

of ''K'' which is obtained by composing an absolutely abelian field with ''K'' and which is

unramified In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...

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 In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

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 \text p \text ;

2^* = -4, 8, -8 \text m \equiv 3 \pmod 4, 2 \pmod 8, -2 \pmod 8 .

Then the genus field is the composite

K(\sqrt).

References

* * * Class field theory {{numtheory-stub

See also

References