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chore(field_theory/splitting_field): split file (#19154)
We split `field_theory.splitting_field` into `field_theory.splitting_field.is_splitting_field` and `field_theory.splitting_field.construction`. This is useful for the port, but also quite a lot of Galois theory should not depend on the existence of splitting fields.
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src/field_theory/splitting_field/is_splitting_field.lean
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| Original file line number | Diff line number | Diff line change |
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| /- | ||
| Copyright (c) 2018 Chris Hughes. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Chris Hughes | ||
| -/ | ||
| import algebra.char_p.algebra | ||
| import field_theory.intermediate_field | ||
| import ring_theory.adjoin.field | ||
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| /-! | ||
| # Splitting fields | ||
| This file introduces the notion of a splitting field of a polynomial and provides an embedding from | ||
| a splitting field to any field that splits the polynomial. A polynomial `f : K[X]` splits | ||
| over a field extension `L` of `K` if it is zero or all of its irreducible factors over `L` have | ||
| degree `1`. A field extension of `K` of a polynomial `f : K[X]` is called a splitting field | ||
| if it is the smallest field extension of `K` such that `f` splits. | ||
| ## Main definitions | ||
| * `polynomial.is_splitting_field`: A predicate on a field to be a splitting field of a polynomial | ||
| `f`. | ||
| ## Main statements | ||
| * `polynomial.is_splitting_field.lift`: An embedding of a splitting field of the polynomial `f` into | ||
| another field such that `f` splits. | ||
| -/ | ||
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| noncomputable theory | ||
| open_locale classical big_operators polynomial | ||
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| universes u v w | ||
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| variables {F : Type u} (K : Type v) (L : Type w) | ||
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| namespace polynomial | ||
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| variables [field K] [field L] [field F] [algebra K L] | ||
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| /-- Typeclass characterising splitting fields. -/ | ||
| class is_splitting_field (f : K[X]) : Prop := | ||
| (splits [] : splits (algebra_map K L) f) | ||
| (adjoin_roots [] : algebra.adjoin K (↑(f.map (algebra_map K L)).roots.to_finset : set L) = ⊤) | ||
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| variables {K L F} | ||
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| namespace is_splitting_field | ||
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| section scalar_tower | ||
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| variables [algebra F K] [algebra F L] [is_scalar_tower F K L] | ||
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| instance map (f : F[X]) [is_splitting_field F L f] : | ||
| is_splitting_field K L (f.map $ algebra_map F K) := | ||
| ⟨by { rw [splits_map_iff, ← is_scalar_tower.algebra_map_eq], exact splits L f }, | ||
| subalgebra.restrict_scalars_injective F $ | ||
| by { rw [map_map, ← is_scalar_tower.algebra_map_eq, subalgebra.restrict_scalars_top, | ||
| eq_top_iff, ← adjoin_roots L f, algebra.adjoin_le_iff], | ||
| exact λ x hx, @algebra.subset_adjoin K _ _ _ _ _ _ hx }⟩ | ||
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| variables (L) | ||
| theorem splits_iff (f : K[X]) [is_splitting_field K L f] : | ||
| polynomial.splits (ring_hom.id K) f ↔ (⊤ : subalgebra K L) = ⊥ := | ||
| ⟨λ h, eq_bot_iff.2 $ adjoin_roots L f ▸ (roots_map (algebra_map K L) h).symm ▸ | ||
| algebra.adjoin_le_iff.2 (λ y hy, | ||
| let ⟨x, hxs, hxy⟩ := finset.mem_image.1 (by rwa multiset.to_finset_map at hy) in | ||
| hxy ▸ set_like.mem_coe.2 $ subalgebra.algebra_map_mem _ _), | ||
| λ h, @ring_equiv.to_ring_hom_refl K _ ▸ | ||
| ring_equiv.self_trans_symm (ring_equiv.of_bijective _ $ algebra.bijective_algebra_map_iff.2 h) ▸ | ||
| by { rw ring_equiv.to_ring_hom_trans, exact splits_comp_of_splits _ _ (splits L f) }⟩ | ||
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| theorem mul (f g : F[X]) (hf : f ≠ 0) (hg : g ≠ 0) [is_splitting_field F K f] | ||
| [is_splitting_field K L (g.map $ algebra_map F K)] : | ||
| is_splitting_field F L (f * g) := | ||
| ⟨(is_scalar_tower.algebra_map_eq F K L).symm ▸ splits_mul _ | ||
| (splits_comp_of_splits _ _ (splits K f)) | ||
| ((splits_map_iff _ _).1 (splits L $ g.map $ algebra_map F K)), | ||
| by rw [polynomial.map_mul, roots_mul (mul_ne_zero (map_ne_zero hf : f.map (algebra_map F L) ≠ 0) | ||
| (map_ne_zero hg)), multiset.to_finset_add, finset.coe_union, | ||
| algebra.adjoin_union_eq_adjoin_adjoin, | ||
| is_scalar_tower.algebra_map_eq F K L, ← map_map, | ||
| roots_map (algebra_map K L) ((splits_id_iff_splits $ algebra_map F K).2 $ splits K f), | ||
| multiset.to_finset_map, finset.coe_image, algebra.adjoin_algebra_map, adjoin_roots, | ||
| algebra.map_top, is_scalar_tower.adjoin_range_to_alg_hom, ← map_map, adjoin_roots, | ||
| subalgebra.restrict_scalars_top]⟩ | ||
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| end scalar_tower | ||
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| variable (L) | ||
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| /-- Splitting field of `f` embeds into any field that splits `f`. -/ | ||
| def lift [algebra K F] (f : K[X]) [is_splitting_field K L f] | ||
| (hf : polynomial.splits (algebra_map K F) f) : L →ₐ[K] F := | ||
| if hf0 : f = 0 then (algebra.of_id K F).comp $ | ||
| (algebra.bot_equiv K L : (⊥ : subalgebra K L) →ₐ[K] K).comp $ | ||
| by { rw ← (splits_iff L f).1 (show f.splits (ring_hom.id K), from hf0.symm ▸ splits_zero _), | ||
| exact algebra.to_top } else | ||
| alg_hom.comp (by { rw ← adjoin_roots L f, exact classical.choice (lift_of_splits _ $ λ y hy, | ||
| have aeval y f = 0, from (eval₂_eq_eval_map _).trans $ | ||
| (mem_roots $ by exact map_ne_zero hf0).1 (multiset.mem_to_finset.mp hy), | ||
| ⟨is_algebraic_iff_is_integral.1 ⟨f, hf0, this⟩, | ||
| splits_of_splits_of_dvd _ hf0 hf $ minpoly.dvd _ _ this⟩) }) | ||
| algebra.to_top | ||
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| theorem finite_dimensional (f : K[X]) [is_splitting_field K L f] : finite_dimensional K L := | ||
| ⟨@algebra.top_to_submodule K L _ _ _ ▸ adjoin_roots L f ▸ | ||
| fg_adjoin_of_finite (finset.finite_to_set _) (λ y hy, | ||
| if hf : f = 0 | ||
| then by { rw [hf, polynomial.map_zero, roots_zero] at hy, cases hy } | ||
| else is_algebraic_iff_is_integral.1 ⟨f, hf, (eval₂_eq_eval_map _).trans $ | ||
| (mem_roots $ by exact map_ne_zero hf).1 (multiset.mem_to_finset.mp hy)⟩)⟩ | ||
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| lemma of_alg_equiv [algebra K F] (p : K[X]) (f : F ≃ₐ[K] L) [is_splitting_field K F p] : | ||
| is_splitting_field K L p := | ||
| begin | ||
| split, | ||
| { rw ← f.to_alg_hom.comp_algebra_map, | ||
| exact splits_comp_of_splits _ _ (splits F p) }, | ||
| { rw [←(algebra.range_top_iff_surjective f.to_alg_hom).mpr f.surjective, | ||
| ←root_set, adjoin_root_set_eq_range (splits F p), root_set, adjoin_roots F p] }, | ||
| end | ||
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| end is_splitting_field | ||
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| end polynomial | ||
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| namespace intermediate_field | ||
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| open polynomial | ||
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| variables {K L} [field K] [field L] [algebra K L] {p : K[X]} | ||
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| lemma splits_of_splits {F : intermediate_field K L} (h : p.splits (algebra_map K L)) | ||
| (hF : ∀ x ∈ p.root_set L, x ∈ F) : p.splits (algebra_map K F) := | ||
| begin | ||
| simp_rw [root_set, finset.mem_coe, multiset.mem_to_finset] at hF, | ||
| rw splits_iff_exists_multiset, | ||
| refine ⟨multiset.pmap subtype.mk _ hF, map_injective _ (algebra_map F L).injective _⟩, | ||
| conv_lhs { rw [polynomial.map_map, ←is_scalar_tower.algebra_map_eq, | ||
| eq_prod_roots_of_splits h, ←multiset.pmap_eq_map _ _ _ hF] }, | ||
| simp_rw [polynomial.map_mul, polynomial.map_multiset_prod, | ||
| multiset.map_pmap, polynomial.map_sub, map_C, map_X], | ||
| refl, | ||
| end | ||
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| end intermediate_field |
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