[Merged by Bors] - feat(field_theory/splitting_field): generalize some results from field to domain #10726
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alexjbest
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@alexjbest I pushed a refactor. By now it might make sense to reorganize properly into a section with variables. |
jcommelin
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Dec 14, 2021
| @@ -422,6 +426,48 @@ calc ((roots ((X : polynomial R) ^ n - C a)).card : with_bot ℕ) | |||
| ≤ degree ((X : polynomial R) ^ n - C a) : card_roots (X_pow_sub_C_ne_zero hn a) | |||
| ... = n : degree_X_pow_sub_C hn a | |||
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| lemma le_root_multiplicity_map {K L : Type*} [comm_ring K] [is_domain K] | |||
| [comm_ring L] [is_domain L] {p : polynomial K} {f : K →+* L} (hf : function.injective f) | |||
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I different version might assume map f p ≠ 0 instead of function.injective f.
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Shall we add a ' version then (or of map_ne_zero)?
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…d to domain (#10726) This PR does a few things generalizing / golfing facts related to polynomials and splitting fields. - Generalize some results in `data.polynomial.field_division` to division rings - generalize `C_leading_coeff_mul_prod_multiset_X_sub_C` from a field to a domain - same for `prod_multiset_X_sub_C_of_monic_of_roots_card_eq` - add a supporting useful lemma `roots_map_of_injective_card_eq_total_degree` saying that if we already have a full (multi)set of roots over a domain, passing along an injection gives the set of roots of the mapped polynomial Inspired by needs of flt-regular. Co-authored-by: Johan Commelin <johan@commelin.net>
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This PR does a few things generalizing / golfing facts related to polynomials and splitting fields.
data.polynomial.field_divisionto division ringsC_leading_coeff_mul_prod_multiset_X_sub_Cfrom a field to a domainprod_multiset_X_sub_C_of_monic_of_roots_card_eqroots_map_of_injective_card_eq_total_degreesaying that if we already have a full (multi)set of roots over a domain, passing along an injection gives the set of roots of the mapped polynomialInspired by needs of flt-regular.