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RamificationGroup/ForMathlib/Algebra/Algebra/PowerBasis.lean
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import Mathlib.RingTheory.PowerBasis | ||
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namespace PowerBasis | ||
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variable {R S : Type*} [CommRing R] [Ring S] [Algebra R S] | ||
{S' : Type*} [Semiring S'] [Algebra R S'] | ||
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theorem algEquiv_ext (pb : PowerBasis R S) {f g : S ≃ₐ[R] S'} (h : f pb.gen = g pb.gen) : | ||
f = g := by | ||
ext x | ||
rw [show f x = g x ↔ f.toAlgHom x = g.toAlgHom x by rfl] | ||
revert x | ||
rw [← AlgHom.ext_iff] | ||
apply algHom_ext _ h | ||
open Algebra PowerBasis Polynomial | ||
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variable {R A : Type*} [CommRing R] [Ring A] [Algebra R A] | ||
{B : Type*} [Semiring B] [Algebra R B] | ||
{F : Type*} [FunLike F A B] [AlgHomClass F R A B] | ||
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theorem Algebra.exists_eq_aeval_generator {s : A} (hgen : adjoin R {s} = ⊤) (x : A) : | ||
∃ f : R[X], x = aeval s f := by | ||
have hx : x ∈ (⊤ : Subalgebra R A) := trivial | ||
rw [← hgen, adjoin_singleton_eq_range_aeval] at hx | ||
rcases hx with ⟨p, hp⟩ | ||
exact ⟨p, hp.symm⟩ | ||
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theorem Algebra.algHomClass_ext_generator {s : A} (hgen : adjoin R {s} = ⊤) | ||
{f g : F} (h : f s = g s) : | ||
f = g := by | ||
apply DFunLike.ext | ||
intro x | ||
have hx : x ∈ (⊤ : Subalgebra R A) := trivial | ||
rw [← hgen, adjoin_singleton_eq_range_aeval] at hx | ||
rcases hx with ⟨p, hp⟩ | ||
simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe] at hp | ||
rw [← hp, ← Polynomial.aeval_algHom_apply, ← Polynomial.aeval_algHom_apply, h] | ||
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theorem PowerBasis.algHom_ext' (pb : PowerBasis R A) {f g : F} (h : f pb.gen = g pb.gen) : | ||
f = g := Algebra.algHomClass_ext_generator (adjoin_gen_eq_top pb) h |
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