- Source: Genus field
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) (m squarefree) is a quadratic field of discriminant D, the genus field of K is a composite of quadratic fields. Let pi run over the prime factors of D. For each such prime p, define p∗ as follows:
p
∗
=
±
p
≡
1
(
mod
4
)
if
p
is odd
;
{\displaystyle p^{*}=\pm p\equiv 1{\pmod {4}}{\text{ if }}p{\text{ is odd}};}
2
∗
=
−
4
,
8
,
−
8
according as
m
≡
3
(
mod
4
)
,
2
(
mod
8
)
,
−
2
(
mod
8
)
.
{\displaystyle 2^{*}=-4,8,-8{\text{ according as }}m\equiv 3{\pmod {4}},2{\pmod {8}},-2{\pmod {8}}.}
Then the genus field is the composite
K
(
p
i
∗
)
.
{\displaystyle K({\sqrt {p_{i}^{*}}}).}
See also
Hilbert class field
References
Ishida, Makoto (1976). The genus fields of algebraic number fields. Lecture Notes in Mathematics. Vol. 555. Springer-Verlag. ISBN 3-540-08000-7. Zbl 0353.12001.
Janusz, Gerald (1973). Algebraic Number Fields. Pure and Applied Mathematics. Vol. 55. Academic Press. ISBN 0-12-380250-4. Zbl 0307.12001.
Lemmermeyer, Franz (2000). Reciprocity laws. From Euler to Eisenstein. Springer Monographs in Mathematics. Berlin: Springer-Verlag. ISBN 3-540-66957-4. MR 1761696. Zbl 0949.11002.
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