fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
@[simp]
rightFDRep [Finite G] : FDRep k G := FDRep.of rightRegular | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | rightFDRep | The right regular representation `rightRegular` on `G → k` as a `FDRep k G`. |
equivHom_injective [Nontrivial k] : Function.Injective (equivHom k G) := by
intro s t h
classical
apply_fun (fun x ↦ (x.hom.hom.app rightFDRep).hom (single t 1) 1) at h
simp_all [single_apply]
@[deprecated (since := "2025-04-27")]
alias equivHom_inj := equivHom_injective | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | equivHom_injective | null |
mulRepHom : rightFDRep (k := k) (G := G) ⊗ rightFDRep ⟶ rightFDRep where
hom := ofHom (LinearMap.mul' k (G → k))
comm := by
intro
ext u
refine TensorProduct.induction_on u rfl (fun _ _ ↦ rfl) (fun _ _ hx hy ↦ ?_)
simp only [map_add, hx, hy] | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | mulRepHom | The `FDRep k G` morphism induced by multiplication on `G → k`. |
map_mul_toRightFDRepComp (η : Aut (forget k G)) (f g : G → k) :
let α : (G → k) →ₗ[k] (G → k) := (η.hom.hom.app rightFDRep).hom
α (f * g) = (α f) * (α g) := by
have nat := η.hom.hom.naturality mulRepHom
have tensor (X Y) : η.hom.hom.app (X ⊗ Y) = (η.hom.hom.app X ⊗ₘ η.hom.hom.app Y) :=
η.hom.isMonoidal.... | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | map_mul_toRightFDRepComp | The `rightFDRep` component of `η : Aut (forget k G)` preserves multiplication |
algHomOfRightFDRepComp (η : Aut (forget k G)) : (G → k) →ₐ[k] (G → k) := by
let α : (G → k) →ₗ[k] (G → k) := (η.hom.hom.app rightFDRep).hom
let α_inv : (G → k) →ₗ[k] (G → k) := (η.inv.hom.app rightFDRep).hom
refine AlgHom.ofLinearMap α ?_ (map_mul_toRightFDRepComp η)
suffices α (α_inv 1) = (1 : G → k) by
ha... | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | algHomOfRightFDRepComp | The `rightFDRep` component of `η : Aut (forget k G)` gives rise to
an algebra morphism `(G → k) →ₐ[k] (G → k)`. |
@[simps]
sumSMulInv [Fintype G] {X : FDRep k G} (v : X) : (G → k) →ₗ[k] X where
toFun f := ∑ s : G, (f s) • (X.ρ s⁻¹ v)
map_add' _ _ := by simp [add_smul, sum_add_distrib]
map_smul' _ _ := by simp [smul_sum, smul_smul]
omit [Finite G] in | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | sumSMulInv | For `v : X` and `G` a finite group, the `G`-equivariant linear map from the right
regular representation `rightFDRep` to `X` sending `single 1 1` to `v`. |
sumSMulInv_single_id [Fintype G] [DecidableEq G] {X : FDRep k G} (v : X) :
∑ s : G, (single 1 1 : G → k) s • (X.ρ s⁻¹) v = v := by
simp | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | sumSMulInv_single_id | null |
@[simps]
ofRightFDRep [Fintype G] (X : FDRep k G) (v : X) : rightFDRep ⟶ X where
hom := ofHom (sumSMulInv v)
comm t := by
ext f
let φ_term (X : FDRep k G) (f : G → k) v s := (f s) • (X.ρ s⁻¹ v)
have := sum_map univ (mulRightEmbedding t⁻¹) (φ_term X (rightRegular t f) v)
simpa [φ_term] using this | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | ofRightFDRep | For `v : X` and `G` a finite group, the representation morphism from the right
regular representation `rightFDRep` to `X` sending `single 1 1` to `v`. |
toRightFDRepComp_injective {η₁ η₂ : Aut (forget k G)}
(h : η₁.hom.hom.app rightFDRep = η₂.hom.hom.app rightFDRep) : η₁ = η₂ := by
have := Fintype.ofFinite G
classical
ext X v
have h1 := η₁.hom.hom.naturality (ofRightFDRep X v)
have h2 := η₂.hom.hom.naturality (ofRightFDRep X v)
rw [h, ← h2] at h1
simp... | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | toRightFDRepComp_injective | null |
leftRegularFDRepHom (s : G) : End (rightFDRep : FDRep k G) where
hom := ofHom (leftRegular s)
comm _ := by
ext f
funext _
apply congrArg f
exact mul_assoc .. | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | leftRegularFDRepHom | `leftRegular` as a morphism `rightFDRep k G ⟶ rightFDRep k G` in `FDRep k G`. |
toRightFDRepComp_in_rightRegular [IsDomain k] (η : Aut (forget k G)) :
∃ (s : G), (η.hom.hom.app rightFDRep).hom = rightRegular s := by
classical
obtain ⟨s, hs⟩ := ((evalAlgHom _ _ 1).comp (algHomOfRightFDRepComp η)).eq_piEvalAlgHom
refine ⟨s, (basisFun k G).ext fun u ↦ ?_⟩
simp only [rightFDRep, forget_obj... | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | toRightFDRepComp_in_rightRegular | null |
equivHom_surjective [IsDomain k] : Function.Surjective (equivHom k G) := by
intro η
obtain ⟨s, h⟩ := toRightFDRepComp_in_rightRegular η
exact ⟨s, toRightFDRepComp_injective (hom_ext h.symm)⟩
variable (k G) in | lemma | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | equivHom_surjective | null |
equiv [IsDomain k] : G ≃* Aut (forget k G) :=
MulEquiv.ofBijective (equivHom k G) ⟨equivHom_injective, equivHom_surjective⟩ | def | RepresentationTheory | [
"Mathlib.RepresentationTheory.FDRep"
] | Mathlib/RepresentationTheory/Tannaka.lean | equiv | Tannaka duality for finite groups:
A finite group `G` is isomorphic to `Aut (forget k G)`, where `k` is any integral domain,
and `forget k G` is the monoidal forgetful functor `FDRep k G ⥤ FGModuleCat k G`. |
AdjoinRoot [CommRing R] (f : R[X]) : Type u :=
Polynomial R ⧸ (span {f} : Ideal R[X]) | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot | Adjoin a root of a polynomial `f` to a commutative ring `R`. We define the new ring
as the quotient of `R[X]` by the principal ideal generated by `f`. |
instCommRing : CommRing (AdjoinRoot f) :=
Ideal.Quotient.commRing _ | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | instCommRing | null |
protected nontrivial [IsDomain R] (h : degree f ≠ 0) : Nontrivial (AdjoinRoot f) :=
Ideal.Quotient.nontrivial
(by
simp_rw [Ne, span_singleton_eq_top, Polynomial.isUnit_iff, not_exists, not_and]
rintro x hx rfl
exact h (degree_C hx.ne_zero)) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | nontrivial | null |
mk : R[X] →+* AdjoinRoot f :=
Ideal.Quotient.mk _
@[elab_as_elim] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk | Ring homomorphism from `R[x]` to `AdjoinRoot f` sending `X` to the `root`. |
induction_on {C : AdjoinRoot f → Prop} (x : AdjoinRoot f) (ih : ∀ p : R[X], C (mk f p)) :
C x :=
Quotient.inductionOn' x ih | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | induction_on | null |
of : R →+* AdjoinRoot f :=
(mk f).comp C | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | of | Embedding of the original ring `R` into `AdjoinRoot f`. |
instSMulAdjoinRoot [DistribSMul S R] [IsScalarTower S R R] : SMul S (AdjoinRoot f) :=
Submodule.Quotient.instSMul' _ | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | instSMulAdjoinRoot | null |
@[simp]
smul_mk [DistribSMul S R] [IsScalarTower S R R] (a : S) (x : R[X]) :
a • mk f x = mk f (a • x) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | smul_mk | null |
smul_of [DistribSMul S R] [IsScalarTower S R R] (a : S) (x : R) :
a • of f x = of f (a • x) := by rw [of, RingHom.comp_apply, RingHom.comp_apply, smul_mk, smul_C] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | smul_of | null |
isScalarTower_right [DistribSMul S R] [IsScalarTower S R R] :
IsScalarTower S (AdjoinRoot f) (AdjoinRoot f) :=
Ideal.Quotient.isScalarTower_right | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | isScalarTower_right | null |
mkₐ : R[X] →ₐ[R] AdjoinRoot f :=
Ideal.Quotient.mkₐ R _
@[simp, norm_cast] theorem mkₐ_toRingHom : ↑(mkₐ f) = mk f := rfl
@[simp] theorem coe_mkₐ : ⇑(mkₐ f) = mk f := rfl
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mkₐ | `R[x]/(f)` is `R`-algebra -/
@[stacks 09FX "second part"]
instance [CommSemiring S] [Algebra S R] : Algebra S (AdjoinRoot f) :=
Ideal.Quotient.algebra S
/- TODO : generalise base ring -/
/-- `R`-algebra homomorphism from `R[x]` to `AdjoinRoot f` sending `X` to the `root`. |
algebraMap_eq : algebraMap R (AdjoinRoot f) = of f :=
rfl
variable (S) in | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | algebraMap_eq | null |
algebraMap_eq' [CommSemiring S] [Algebra S R] :
algebraMap S (AdjoinRoot f) = (of f).comp (algebraMap S R) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | algebraMap_eq' | null |
finiteType : Algebra.FiniteType R (AdjoinRoot f) :=
.of_surjective _ (Ideal.Quotient.mkₐ_surjective R _) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | finiteType | null |
finitePresentation : Algebra.FinitePresentation R (AdjoinRoot f) :=
(Algebra.FinitePresentation.polynomial R).quotient (Submodule.fg_span_singleton f) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | finitePresentation | null |
root : AdjoinRoot f :=
mk f X
variable {f} | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | root | The adjoined root. |
hasCoeT : CoeTC R (AdjoinRoot f) :=
⟨of f⟩ | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | hasCoeT | null |
@[ext]
algHom_ext [Semiring S] [Algebra R S] {g₁ g₂ : AdjoinRoot f →ₐ[R] S}
(h : g₁ (root f) = g₂ (root f)) : g₁ = g₂ :=
Ideal.Quotient.algHom_ext R <| Polynomial.algHom_ext h
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | algHom_ext | Two `R`-`AlgHom` from `AdjoinRoot f` to the same `R`-algebra are the same iff
they agree on `root f`. |
mk_eq_mk {g h : R[X]} : mk f g = mk f h ↔ f ∣ g - h :=
Ideal.Quotient.eq.trans Ideal.mem_span_singleton
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk_eq_mk | null |
mk_eq_zero {g : R[X]} : mk f g = 0 ↔ f ∣ g :=
mk_eq_mk.trans <| by rw [sub_zero]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk_eq_zero | null |
mk_self : mk f f = 0 :=
Quotient.sound' <| QuotientAddGroup.leftRel_apply.mpr (mem_span_singleton.2 <| by simp)
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk_self | null |
mk_C (x : R) : mk f (C x) = x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk_C | null |
mk_X : mk f X = root f :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk_X | null |
mk_ne_zero_of_degree_lt (hf : Monic f) {g : R[X]} (h0 : g ≠ 0) (hd : degree g < degree f) :
mk f g ≠ 0 :=
mk_eq_zero.not.2 <| hf.not_dvd_of_degree_lt h0 hd | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk_ne_zero_of_degree_lt | null |
mk_ne_zero_of_natDegree_lt (hf : Monic f) {g : R[X]} (h0 : g ≠ 0)
(hd : natDegree g < natDegree f) : mk f g ≠ 0 :=
mk_eq_zero.not.2 <| hf.not_dvd_of_natDegree_lt h0 hd
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk_ne_zero_of_natDegree_lt | null |
aeval_eq (p : R[X]) : aeval (root f) p = mk f p :=
Polynomial.induction_on p
(fun x => by
rw [aeval_C]
rfl)
(fun p q ihp ihq => by rw [map_add, RingHom.map_add, ihp, ihq]) fun n x _ => by
rw [map_mul, aeval_C, map_pow, aeval_X, RingHom.map_mul, mk_C, RingHom.map_pow, mk_X]
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | aeval_eq | null |
adjoinRoot_eq_top : Algebra.adjoin R ({root f} : Set (AdjoinRoot f)) = ⊤ := by
refine Algebra.eq_top_iff.2 fun x => ?_
induction x using AdjoinRoot.induction_on with
| ih p => exact (Algebra.adjoin_singleton_eq_range_aeval R (root f)).symm ▸ ⟨p, aeval_eq p⟩
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | adjoinRoot_eq_top | null |
eval₂_root (f : R[X]) : f.eval₂ (of f) (root f) = 0 := by
rw [← algebraMap_eq, ← aeval_def, aeval_eq, mk_self] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | eval₂_root | null |
isRoot_root (f : R[X]) : IsRoot (f.map (of f)) (root f) := by
rw [IsRoot, eval_map, eval₂_root] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | isRoot_root | null |
isAlgebraic_root (hf : f ≠ 0) : IsAlgebraic R (root f) :=
⟨f, hf, eval₂_root f⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | isAlgebraic_root | null |
of.injective_of_degree_ne_zero [IsDomain R] (hf : f.degree ≠ 0) :
Function.Injective (AdjoinRoot.of f) := by
rw [injective_iff_map_eq_zero]
intro p hp
rw [AdjoinRoot.of, RingHom.comp_apply, AdjoinRoot.mk_eq_zero] at hp
by_cases h : f = 0
· exact C_eq_zero.mp (eq_zero_of_zero_dvd (by rwa [h] at hp))
· co... | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | of.injective_of_degree_ne_zero | null |
lift (i : R →+* S) (x : S) (h : f.eval₂ i x = 0) : AdjoinRoot f →+* S := by
apply Ideal.Quotient.lift _ (eval₂RingHom i x)
intro g H
rcases mem_span_singleton.1 H with ⟨y, hy⟩
rw [hy, RingHom.map_mul, coe_eval₂RingHom, h, zero_mul]
variable {i : R →+* S} {a : S} (h : f.eval₂ i a = 0)
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | lift | Lift a ring homomorphism `i : R →+* S` to `AdjoinRoot f →+* S`. |
lift_mk (g : R[X]) : lift i a h (mk f g) = g.eval₂ i a :=
Ideal.Quotient.lift_mk _ _ _
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | lift_mk | null |
lift_root : lift i a h (root f) = a := by rw [root, lift_mk, eval₂_X]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | lift_root | null |
lift_of {x : R} : lift i a h x = i x := by rw [← mk_C x, lift_mk, eval₂_C]
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | lift_of | null |
lift_comp_of : (lift i a h).comp (of f) = i :=
RingHom.ext fun _ => @lift_of _ _ _ _ _ _ _ h _
variable (f) [Algebra R S] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | lift_comp_of | null |
liftHom (x : S) (hfx : aeval x f = 0) : AdjoinRoot f →ₐ[R] S :=
{ lift (algebraMap R S) x hfx with
commutes' := fun r => show lift _ _ hfx r = _ from lift_of hfx }
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | liftHom | Produce an algebra homomorphism `AdjoinRoot f →ₐ[R] S` sending `root f` to
a root of `f` in `S`. |
coe_liftHom (x : S) (hfx : aeval x f = 0) :
(liftHom f x hfx : AdjoinRoot f →+* S) = lift (algebraMap R S) x hfx :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | coe_liftHom | null |
aeval_algHom_eq_zero (ϕ : AdjoinRoot f →ₐ[R] S) : aeval (ϕ (root f)) f = 0 := by
have h : ϕ.toRingHom.comp (of f) = algebraMap R S := RingHom.ext_iff.mpr ϕ.commutes
rw [aeval_def, ← h, ← RingHom.map_zero ϕ.toRingHom, ← eval₂_root f, hom_eval₂]
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | aeval_algHom_eq_zero | null |
liftHom_eq_algHom (f : R[X]) (ϕ : AdjoinRoot f →ₐ[R] S) :
liftHom f (ϕ (root f)) (aeval_algHom_eq_zero f ϕ) = ϕ := by
suffices AlgHom.equalizer ϕ (liftHom f (ϕ (root f)) (aeval_algHom_eq_zero f ϕ)) = ⊤ by
exact (AlgHom.ext fun x => (SetLike.ext_iff.mp this x).mpr Algebra.mem_top).symm
rw [eq_top_iff, ← adjo... | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | liftHom_eq_algHom | null |
liftHom_mk {g : R[X]} : liftHom f a hfx (mk f g) = aeval a g :=
lift_mk hfx g
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | liftHom_mk | null |
liftHom_root : liftHom f a hfx (root f) = a :=
lift_root hfx
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | liftHom_root | null |
liftHom_of {x : R} : liftHom f a hfx (of f x) = algebraMap _ _ x :=
lift_of hfx | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | liftHom_of | null |
@[simp]
root_isInv (r : R) : of _ r * root (C r * X - 1) = 1 := by
convert sub_eq_zero.1 ((eval₂_sub _).symm.trans <| eval₂_root <| C r * X - 1) <;>
simp only [eval₂_mul, eval₂_C, eval₂_X, eval₂_one] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | root_isInv | null |
algHom_subsingleton {S : Type*} [CommRing S] [Algebra R S] {r : R} :
Subsingleton (AdjoinRoot (C r * X - 1) →ₐ[R] S) :=
⟨fun f g =>
algHom_ext
(@inv_unique _ _ (algebraMap R S r) _ _
(by rw [← f.commutes, ← map_mul, algebraMap_eq, root_isInv, map_one])
(by rw [← g.commutes, ← map_mul, al... | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | algHom_subsingleton | null |
isDomain_of_prime (hf : Prime f) : IsDomain (AdjoinRoot f) :=
(Ideal.Quotient.isDomain_iff_prime (span {f} : Ideal R[X])).mpr <|
(Ideal.span_singleton_prime hf.ne_zero).mpr hf | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | isDomain_of_prime | null |
noZeroSMulDivisors_of_prime_of_degree_ne_zero [IsDomain R] (hf : Prime f)
(hf' : f.degree ≠ 0) : NoZeroSMulDivisors R (AdjoinRoot f) :=
haveI := isDomain_of_prime hf
NoZeroSMulDivisors.iff_algebraMap_injective.mpr (of.injective_of_degree_ne_zero hf') | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | noZeroSMulDivisors_of_prime_of_degree_ne_zero | null |
span_maximal_of_irreducible [Fact (Irreducible f)] : (span {f}).IsMaximal :=
PrincipalIdealRing.isMaximal_of_irreducible <| Fact.out | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | span_maximal_of_irreducible | null |
noncomputable instGroupWithZero [Fact (Irreducible f)] : GroupWithZero (AdjoinRoot f) :=
Quotient.groupWithZero (span {f} : Ideal K[X]) | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | instGroupWithZero | null |
@[stacks 09FX "first part, see also 09FI"]
noncomputable instField [Fact (Irreducible f)] : Field (AdjoinRoot f) where
__ := instCommRing _
__ := instGroupWithZero
nnqsmul := (· • ·)
qsmul := (· • ·)
nnratCast_def q := by
rw [← map_natCast (of f), ← map_natCast (of f), ← map_div₀, ← NNRat.cast_def]; rfl
... | instance | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | instField | If `R` is a field and `f` is irreducible, then `AdjoinRoot f` is a field |
coe_injective (h : degree f ≠ 0) : Function.Injective ((↑) : K → AdjoinRoot f) :=
have := AdjoinRoot.nontrivial f h
(of f).injective | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | coe_injective | null |
coe_injective' [Fact (Irreducible f)] : Function.Injective ((↑) : K → AdjoinRoot f) :=
(of f).injective
variable (f) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | coe_injective' | null |
mul_div_root_cancel [Fact (Irreducible f)] :
(X - C (root f)) * ((f.map (of f)) / (X - C (root f))) = f.map (of f) :=
mul_div_eq_iff_isRoot.2 <| isRoot_root _ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mul_div_root_cancel | null |
isIntegral_root' (hg : g.Monic) : IsIntegral R (root g) :=
⟨g, hg, eval₂_root g⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | isIntegral_root' | null |
modByMonicHom (hg : g.Monic) : AdjoinRoot g →ₗ[R] R[X] :=
(Submodule.liftQ _ (Polynomial.modByMonicHom g)
fun f (hf : f ∈ (Ideal.span {g}).restrictScalars R) =>
(mem_ker_modByMonic hg).mpr (Ideal.mem_span_singleton.mp hf)).comp <|
(Submodule.Quotient.restrictScalarsEquiv R (Ideal.span {g} : Ideal ... | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | modByMonicHom | `AdjoinRoot.modByMonicHom` sends the equivalence class of `f` mod `g` to `f %ₘ g`.
This is a well-defined right inverse to `AdjoinRoot.mk`, see `AdjoinRoot.mk_leftInverse`. |
modByMonicHom_mk (hg : g.Monic) (f : R[X]) : modByMonicHom hg (mk g f) = f %ₘ g :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | modByMonicHom_mk | null |
mk_leftInverse (hg : g.Monic) : Function.LeftInverse (mk g) (modByMonicHom hg) := by
intro f
induction f using AdjoinRoot.induction_on
rw [modByMonicHom_mk hg, mk_eq_mk, modByMonic_eq_sub_mul_div _ hg, sub_sub_cancel_left,
dvd_neg]
apply dvd_mul_right | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk_leftInverse | null |
mk_surjective : Function.Surjective (mk g) :=
Ideal.Quotient.mk_surjective | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | mk_surjective | null |
powerBasisAux' (hg : g.Monic) : Basis (Fin g.natDegree) R (AdjoinRoot g) :=
.ofEquivFun
{ toFun := fun f i => (modByMonicHom hg f).coeff i
invFun := fun c => mk g <| ∑ i : Fin g.natDegree, monomial i (c i)
map_add' := fun f₁ f₂ =>
funext fun i => by simp only [(modByMonicHom hg).map_add, coeff... | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | powerBasisAux' | The elements `1, root g, ..., root g ^ (d - 1)` form a basis for `AdjoinRoot g`,
where `g` is a monic polynomial of degree `d`. |
powerBasisAux'_repr_symm_apply (hg : g.Monic) (c : Fin g.natDegree →₀ R) :
(powerBasisAux' hg).repr.symm c = mk g (∑ i : Fin _, monomial i (c i)) :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | powerBasisAux'_repr_symm_apply | null |
powerBasisAux'_repr_apply_to_fun (hg : g.Monic) (f : AdjoinRoot g) (i : Fin g.natDegree) :
(powerBasisAux' hg).repr f i = (modByMonicHom hg f).coeff ↑i :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | powerBasisAux'_repr_apply_to_fun | null |
@[simps]
powerBasis' (hg : g.Monic) : PowerBasis R (AdjoinRoot g) where
gen := root g
dim := g.natDegree
basis := powerBasisAux' hg
basis_eq_pow i := by
simp only [powerBasisAux', Basis.coe_ofEquivFun, LinearEquiv.coe_symm_mk]
rw [Finset.sum_eq_single i]
· rw [Pi.single_eq_same, monomial_one_right_e... | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | powerBasis' | The power basis `1, root g, ..., root g ^ (d - 1)` for `AdjoinRoot g`,
where `g` is a monic polynomial of degree `d`. |
_root_.Polynomial.Monic.free_adjoinRoot (hg : g.Monic) : Module.Free R (AdjoinRoot g) :=
.of_basis (powerBasis' hg).basis | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | _root_.Polynomial.Monic.free_adjoinRoot | null |
_root_.Polynomial.Monic.finite_adjoinRoot (hg : g.Monic) : Module.Finite R (AdjoinRoot g) :=
.of_basis (powerBasis' hg).basis | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | _root_.Polynomial.Monic.finite_adjoinRoot | null |
_root_.Polynomial.Monic.free_quotient (hg : g.Monic) :
Module.Free R (R[X] ⧸ Ideal.span {g}) :=
hg.free_adjoinRoot | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | _root_.Polynomial.Monic.free_quotient | An unwrapped version of `AdjoinRoot.free_of_monic` for better discoverability. |
_root_.Polynomial.Monic.finite_quotient (hg : g.Monic) :
Module.Finite R (R[X] ⧸ Ideal.span {g}) :=
hg.finite_adjoinRoot
variable [Field K] {f : K[X]} | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | _root_.Polynomial.Monic.finite_quotient | An unwrapped version of `AdjoinRoot.finite_of_monic` for better discoverability. |
isIntegral_root (hf : f ≠ 0) : IsIntegral K (root f) :=
(isAlgebraic_root hf).isIntegral | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | isIntegral_root | null |
minpoly_root (hf : f ≠ 0) : minpoly K (root f) = f * C f.leadingCoeff⁻¹ := by
have f'_monic : Monic _ := monic_mul_leadingCoeff_inv hf
refine (minpoly.unique K _ f'_monic ?_ ?_).symm
· rw [map_mul, aeval_eq, mk_self, zero_mul]
intro q q_monic q_aeval
have commutes : (lift (algebraMap K (AdjoinRoot f)) (root f... | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | minpoly_root | null |
powerBasisAux (hf : f ≠ 0) : Basis (Fin f.natDegree) K (AdjoinRoot f) := by
let f' := f * C f.leadingCoeff⁻¹
have deg_f' : f'.natDegree = f.natDegree := by
rw [natDegree_mul hf, natDegree_C, add_zero]
· rwa [Ne, C_eq_zero, inv_eq_zero, leadingCoeff_eq_zero]
have minpoly_eq : minpoly K (root f) = f' := min... | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | powerBasisAux | The elements `1, root f, ..., root f ^ (d - 1)` form a basis for `AdjoinRoot f`,
where `f` is an irreducible polynomial over a field of degree `d`. |
@[simps!]
powerBasis (hf : f ≠ 0) : PowerBasis K (AdjoinRoot f) where
gen := root f
dim := f.natDegree
basis := powerBasisAux hf
basis_eq_pow := by simp [powerBasisAux] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | powerBasis | The power basis `1, root f, ..., root f ^ (d - 1)` for `AdjoinRoot f`,
where `f` is an irreducible polynomial over a field of degree `d`. |
minpoly_powerBasis_gen (hf : f ≠ 0) :
minpoly K (powerBasis hf).gen = f * C f.leadingCoeff⁻¹ := by
rw [powerBasis_gen, minpoly_root hf] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | minpoly_powerBasis_gen | null |
minpoly_powerBasis_gen_of_monic (hf : f.Monic) (hf' : f ≠ 0 := hf.ne_zero) :
minpoly K (powerBasis hf').gen = f := by
rw [minpoly_powerBasis_gen hf', hf.leadingCoeff, inv_one, C.map_one, mul_one] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | minpoly_powerBasis_gen_of_monic | null |
_root_.finrank_quotient_span_eq_natDegree {f : K[X]} :
Module.finrank K (K[X] ⧸ Ideal.span {f}) = f.natDegree := by
by_cases hf : f = 0
· rw [hf, natDegree_zero,
((Submodule.quotEquivOfEqBot _ (by simp)).restrictScalars K).finrank_eq]
exact finrank_of_not_finite Polynomial.not_finite
rw [PowerBasis.... | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | _root_.finrank_quotient_span_eq_natDegree | See `finrank_quotient_span_eq_natDegree'` for a version over a ring when `f` is monic. |
Minpoly.toAdjoin : AdjoinRoot (minpoly R x) →ₐ[R] adjoin R ({x} : Set S) :=
liftHom _ ⟨x, self_mem_adjoin_singleton R x⟩
(by simp [← Subalgebra.coe_eq_zero, aeval_subalgebra_coe])
variable {R x}
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | Minpoly.toAdjoin | The surjective algebra morphism `R[X]/(minpoly R x) → R[x]`.
If `R` is a integrally closed domain and `x` is integral, this is an isomorphism,
see `minpoly.equivAdjoin`. |
Minpoly.coe_toAdjoin :
⇑(Minpoly.toAdjoin R x) = liftHom (minpoly R x) ⟨x, self_mem_adjoin_singleton R x⟩
(by simp [← Subalgebra.coe_eq_zero, aeval_subalgebra_coe]) := rfl
@[deprecated (since := "2025-07-21")] alias Minpoly.toAdjoin_apply := Minpoly.coe_toAdjoin
@[deprecated (since := "2025-07-21")] alias Min... | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | Minpoly.coe_toAdjoin | null |
Minpoly.coe_toAdjoin_mk_X : Minpoly.toAdjoin R x (mk (minpoly R x) X) = x := by simp
@[deprecated (since := "2025-07-21")] alias Minpoly.toAdjoin.apply_X := Minpoly.coe_toAdjoin_mk_X
variable (R x) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | Minpoly.coe_toAdjoin_mk_X | null |
Minpoly.toAdjoin.surjective : Function.Surjective (Minpoly.toAdjoin R x) := by
rw [← AlgHom.range_eq_top, _root_.eq_top_iff, ← adjoin_adjoin_coe_preimage]
exact adjoin_le fun ⟨y₁, y₂⟩ h ↦ ⟨mk (minpoly R x) X, by simpa using h.symm⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | Minpoly.toAdjoin.surjective | null |
@[simps -fullyApplied]
equiv' (h₁ : aeval (root g) (minpoly R pb.gen) = 0) (h₂ : aeval pb.gen g = 0) :
AdjoinRoot g ≃ₐ[R] S :=
{ AdjoinRoot.liftHom g pb.gen h₂ with
toFun := AdjoinRoot.liftHom g pb.gen h₂
invFun := pb.lift (root g) h₁
left_inv x := AdjoinRoot.induction_on _ x fun x => by rw [liftHom_m... | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | equiv' | If `S` is an extension of `R` with power basis `pb` and `g` is a monic polynomial over `R`
such that `pb.gen` has a minimal polynomial `g`, then `S` is isomorphic to `AdjoinRoot g`.
Compare `PowerBasis.equivOfRoot`, which would require
`h₂ : aeval pb.gen (minpoly R (root g)) = 0`; that minimal polynomial is not
guaran... |
equiv'_toAlgHom (h₁ : aeval (root g) (minpoly R pb.gen) = 0) (h₂ : aeval pb.gen g = 0) :
(equiv' g pb h₁ h₂).toAlgHom = AdjoinRoot.liftHom g pb.gen h₂ :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | equiv'_toAlgHom | null |
equiv'_symm_toAlgHom (h₁ : aeval (root g) (minpoly R pb.gen) = 0)
(h₂ : aeval pb.gen g = 0) : (equiv' g pb h₁ h₂).symm.toAlgHom = pb.lift (root g) h₁ :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | equiv'_symm_toAlgHom | null |
equiv (f : F[X]) (hf : f ≠ 0) :
(AdjoinRoot f →ₐ[F] L) ≃ { x // x ∈ f.aroots L } :=
(powerBasis hf).liftEquiv'.trans
((Equiv.refl _).subtypeEquiv fun x => by
rw [powerBasis_gen, minpoly_root hf, aroots_mul, aroots_C, add_zero, Equiv.refl_apply]
exact (monic_mul_leadingCoeff_inv hf).ne_zero) | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | equiv | If `L` is a field extension of `F` and `f` is a polynomial over `F` then the set
of maps from `F[x]/(f)` into `L` is in bijection with the set of roots of `f` in `L`. |
quotMapOfEquivQuotMapCMapSpanMk :
AdjoinRoot f ⧸ I.map (of f) ≃+*
AdjoinRoot f ⧸ (I.map (C : R →+* R[X])).map (Ideal.Quotient.mk (span {f})) :=
Ideal.quotEquivOfEq (by rw [of, AdjoinRoot.mk, Ideal.map_map])
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | quotMapOfEquivQuotMapCMapSpanMk | The natural isomorphism `R[α]/(I[α]) ≅ R[α]/((I[x] ⊔ (f)) / (f))` for `α` a root of
`f : R[X]` and `I : Ideal R`.
See `adjoin_root.quot_map_of_equiv` for the isomorphism with `(R/I)[X] / (f mod I)`. |
quotMapOfEquivQuotMapCMapSpanMk_mk (x : AdjoinRoot f) :
quotMapOfEquivQuotMapCMapSpanMk I f (Ideal.Quotient.mk (I.map (of f)) x) =
Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk (span {f})) (I.map (C : R →+* R[X]))) x := rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | quotMapOfEquivQuotMapCMapSpanMk_mk | null |
quotMapOfEquivQuotMapCMapSpanMk_symm_mk (x : AdjoinRoot f) :
(quotMapOfEquivQuotMapCMapSpanMk I f).symm
(Ideal.Quotient.mk ((I.map (C : R →+* R[X])).map (Ideal.Quotient.mk (span {f}))) x) =
Ideal.Quotient.mk (I.map (of f)) x := by
rw [quotMapOfEquivQuotMapCMapSpanMk, Ideal.quotEquivOfEq_symm]
exac... | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | quotMapOfEquivQuotMapCMapSpanMk_symm_mk | null |
quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk :
AdjoinRoot f ⧸ (I.map (C : R →+* R[X])).map (Ideal.Quotient.mk (span ({f} : Set R[X]))) ≃+*
(R[X] ⧸ I.map (C : R →+* R[X])) ⧸
(span ({f} : Set R[X])).map (Ideal.Quotient.mk (I.map (C : R →+* R[X]))) :=
quotQuotEquivComm (Ideal.span ({f} : Set R[X])) (I.m... | def | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk | The natural isomorphism `R[α]/((I[x] ⊔ (f)) / (f)) ≅ (R[x]/I[x])/((f) ⊔ I[x] / I[x])`
for `α` a root of `f : R[X]` and `I : Ideal R` |
quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_mk (p : R[X]) :
quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk I f (Ideal.Quotient.mk _ (mk f p)) =
quotQuotMk (I.map C) (span {f}) p :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_mk | null |
quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_symm_quotQuotMk (p : R[X]) :
(quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk I f).symm (quotQuotMk (I.map C) (span {f}) p) =
Ideal.Quotient.mk (Ideal.map (Ideal.Quotient.mk (span {f})) (I.map (C : R →+* R[X])))
(mk f p) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Defs",
"Mathlib.Algebra.Polynomial.FieldDivision",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.RingTheory.FinitePresentation",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Ideal.Quotient.Noetherian",
"Mathlib.RingTheory.PowerBasis",... | Mathlib/RingTheory/AdjoinRoot.lean | quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_symm_quotQuotMk | null |
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