Conjugate fields of a polynomial root #
Let x lie in a normal extension of F, and put N for the normal closure of F⟮x⟯ in
that extension. The simple field F⟮x⟯ embeds in N; this file records its orbit under
Gal(N/F). These are the distinct conjugate fields generated by the roots of minpoly F x.
When the minimal polynomial is separable, the number of these fields is
[Gal(N/F) : N_G(H)],
where H is the point stabilizer in the polynomial Galois group, transported to Gal(N/F) by
TauCeti.galEquivNormalClosure. Thus conjugate fields are indexed by the normalizer quotient,
not by the generally larger root quotient G / H, which indexes the embeddings of F⟮x⟯
(see TauCeti.FieldTheory.GaloisGroups.Embeddings).
Main definitions #
TauCeti.conjugateSimpleFields: its set of conjugates underGal(N/F).TauCeti.quotientNormalizerEquivConjugateSimpleFields: the normalizer quotient indexing those fields.TauCeti.quotientGalNormalizerEquivConjugateSimpleFields: the same indexing through the polynomial Galois group.
Main results #
TauCeti.quotientGalStabilizerEquivAlgHomSimpleField_fieldRange: the embedding indexed by a root-stabilizer coset has as image the conjugate field indexed by its projection to the normalizer quotient.TauCeti.conjugateSimpleFieldsEquivConjugateSubgroups: the conjugate fields correspond to conjugates of a point stabilizer in the polynomial Galois group.TauCeti.ncard_conjugateSimpleFields: the number of conjugate simple fields is the index of the normalizer of their fixing subgroup.TauCeti.exists_root_ncard_conjugateSimpleFields_eq_index_normalizer: the same formula with that fixing subgroup identified as a transported point stabilizer ofminpoly F x.
The distinct conjugates of F⟮x⟯ inside its normal closure.
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A field is conjugate to F⟮x⟯ exactly when it is the image of that field under an
automorphism of its normal closure.
The conjugate simple fields are precisely the fields generated by roots of the minimal polynomial inside the normal closure.
For a root whose point stabilizer fixes F⟮x⟯, conjugates of the simple field correspond
to conjugates of that stabilizer in the polynomial Galois group. Such a root exists by
exists_root_map_stabilizer_eq_fixingSubgroup.
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The polynomial-group correspondence sends a conjugate simple field to the inverse image of its fixing subgroup under the normal-closure Galois-group equivalence.
The fixing subgroup of the field corresponding to a conjugate point stabilizer is its transported subgroup.
The field corresponding to a conjugate point stabilizer is the fixed field of its transported subgroup.
A normalizer coset for the simple field specifies one of its conjugate images.
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A coset representative sends the simple field to its image under that automorphism.
The normalizer-coset parametrization of simple fields is Galois-equivariant.
Cosets of the normalizer of a point stabilizer in the polynomial Galois group index the conjugate images of the simple field.
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- One or more equations did not get rendered due to their size.
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A polynomial Galois coset represented by σ gives the field obtained by the
corresponding automorphism of the normal closure.
The field range of the embedding indexed by a root-stabilizer coset is the conjugate field indexed by the image of that coset in the normalizer quotient.
The polynomial normalizer-coset parametrization respects the transported Galois action.
For a separable minimal polynomial, the number of conjugates of F⟮x⟯ is the index of
the normalizer of its fixing subgroup in the Galois group of the normal closure.
The number of conjugate simple fields is the index of the normalizer of a point stabilizer in the polynomial Galois group, when that stabilizer fixes the original field.
A root corresponding to the original generator identifies the fixing subgroup of F⟮x⟯
with its point stabilizer. Hence the number of conjugate fields is the index of the normalizer
of that transported stabilizer.