The genus field of a quadratic field has Galois group Cl⁺(K)/Cl⁺(K)² #
For a squarefree integer d let K = ℚ(√d) be the embedded quadratic base of the candidate genus
field K_gen = candidateGenusField hd, the compositum of the quadratic fields of the prime
discriminants dividing disc K. It is the narrow genus field of K
(isNarrowGenusField_candidateGenusField), and the genus field of K when d < 0
(isGenusField_candidateGenusField). This file proves the genus-field isomorphism
Gal(K_gen / K) ≅ Cl⁺(K) / Cl⁺(K)²,
and, for imaginary K, its ordinary form Gal(K_gen / K) ≅ Cl(K) / Cl(K)².
Both sides are described by the same space of sign patterns on the prime discriminants, and the
isomorphism is the identification of the two descriptions. On the Galois side, an automorphism is
recorded by the signs it puts on the chosen square roots, and fixing the base is exactly having an
even number of sign changes (galoisGroupEquivCandidateGenusFieldRelative). On the class-group
side, a narrow ideal class is recorded by its genus characters, which realize exactly the sign
patterns of product one and separate the classes modulo squares
(mem_range_genusCharFunElementaryTwoQuotientFamilyLinearMap_iff and
genusCharFunElementaryTwoQuotientFamilyLinearMap_injective). The two sign spaces coincide, so
the two groups do.
The composite is the inverse of the Artin map of K_gen / K: the genus character χ_P of the
class of a degree-one prime 𝔮 above q is the Legendre symbol governing the Frobenius of q in
ℚ(√P) (exists_forall_genusCharFunNarrowClassGroupHom_eq). That identification with Frobenius
elements is not proved here; what is proved is the isomorphism itself.
See D. A. Cox, Primes of the Form x² + ny², §6.A, and F. Lemmermeyer, Reciprocity Laws: From Euler to Eisenstein, §2.2.
Main definitions #
TauCeti.Multiquadratic.narrowElementaryTwoQuotientEquivRelativeSign:Cl⁺(K)/Cl⁺(K)²is the space of even-parity sign patterns on the prime discriminants ofdisc K.TauCeti.Multiquadratic.autCandidateGenusFieldEquivNarrowElementaryTwoQuotient:Gal(K_gen/K) ≅ Cl⁺(K)/Cl⁺(K)².TauCeti.Multiquadratic.autCandidateGenusFieldEquivElementaryTwoQuotient: ford < 0,Gal(K_gen/K) ≅ Cl(K)/Cl(K)².
The genus characters of the embedded base, in sign coordinates #
The genus characters of the embedded quadratic base ℚ(√d), assembled into a ZMod 2-linear
map into the sign patterns on the prime discriminants of its discriminant. This is
genusCharFunElementaryTwoQuotientFamilyLinearMap for the chosen factorization
genusPrimeDiscriminants hd, with the sign group ℤˣ written as ZMod 2 so that the target is
the one carrying the relative Galois group.
Instances For
The sign pattern recorded by the genus characters is the image of the sign vector of
genusCharFunElementaryTwoQuotientFamilyLinearMap under the identification of ℤˣ with
ZMod 2.
The genus characters of the base realize exactly the even-parity sign patterns.
The genus characters of the base separate narrow classes modulo squares.
The genus-field isomorphism #
Cl⁺(K)/Cl⁺(K)² is the even-parity sign space. For K = ℚ(√d) the embedded quadratic base
of the candidate genus field, the genus characters identify the maximal elementary-2 quotient of
the narrow class group with the 𝔽₂-space of sign patterns of product one on the prime
discriminants dividing disc K.
Instances For
The sign vector attached to a narrow class modulo squares is the vector of its genus
characters: narrowElementaryTwoQuotientEquivRelativeSign is the genus-character map.
The genus-field isomorphism Gal(K_gen/K) ≅ Cl⁺(K)/Cl⁺(K)². For a squarefree integer d
that is not a rational square, the Galois group of the candidate genus field over its embedded
quadratic base K = ℚ(√d) is isomorphic to the maximal elementary-2 quotient of the narrow class
group of K. Both are the space of even-parity sign patterns on the prime discriminants dividing
disc K: the Galois group by its sign patterns, the class group by its genus characters.
The map is pinned down coordinate for coordinate on those prime discriminants: it sends an
automorphism σ to the unique narrow class modulo squares whose genus characters are the signs
that σ puts on the chosen square roots
(autCandidateGenusFieldEquivNarrowElementaryTwoQuotient_apply together with
narrowElementaryTwoQuotientEquivRelativeSign_apply_coe). Its identification with the inverse
Artin map of K_gen / K is not proved here; see the module docstring.
Instances For
The genus-field isomorphism sends an automorphism to the narrow class modulo squares that the genus characters attach to its sign pattern.
The genus-field isomorphism for an imaginary quadratic field,
Gal(K_gen/K) ≅ Cl(K)/Cl(K)². For d < 0 squarefree, the narrow and ordinary class groups of
K = ℚ(√d) agree, so the genus-field isomorphism takes its classical ordinary form. Here
K_gen is the genus field of K in the full sense
(isGenusField_candidateGenusField): unramified over K at every place, including the infinite
ones, and abelian over ℚ.
Instances For
The imaginary genus-field isomorphism is the narrow genus-field isomorphism followed by the identification of the narrow and ordinary class-group quotients.