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 ⌀ |
|---|---|---|---|---|---|---|
AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_none
(hx : AlgebraicIndependent R x) :
hx.mvPolynomialOptionEquivPolynomialAdjoin (X none) = Polynomial.X := by
rw [AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_apply, aeval_X, Option.elim,
Polynomial.map_X] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.Algebra.MvPolynomial.Monad",
"Mathlib.Algebra.MvPolynomial.Supported",
"Mathlib.RingTheory.AlgebraicIndependent.Defs",
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.RingTheory.MvPolynomial.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Basic.lean | AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_none | null |
AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_some
(hx : AlgebraicIndependent R x) (i) :
hx.mvPolynomialOptionEquivPolynomialAdjoin (X (some i)) =
Polynomial.C (hx.aevalEquiv (X i)) := by
rw [AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_apply, aeval_X, Option.elim,
Polynomial.map_C, RingHom.coe_coe] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.Algebra.MvPolynomial.Monad",
"Mathlib.Algebra.MvPolynomial.Supported",
"Mathlib.RingTheory.AlgebraicIndependent.Defs",
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.RingTheory.MvPolynomial.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Basic.lean | AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_some | null |
AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin
(hx : AlgebraicIndependent R x) (a : A) :
RingHom.comp
(↑(Polynomial.aeval a : Polynomial (adjoin R (Set.range x)) →ₐ[_] A) :
Polynomial (adjoin R (Set.range x)) →+* A)
hx.mvPolynomialOptionEquivPolynomialAdjoin.toRingHom =
↑(MvPolynomial.aeval fun o : Option ι => o.elim a x : MvPolynomial (Option ι) R →ₐ[R] A) := by
refine MvPolynomial.ringHom_ext ?_ ?_ <;>
simp only [RingHom.comp_apply, RingEquiv.toRingHom_eq_coe, RingEquiv.coe_toRingHom,
AlgHom.coe_toRingHom, AlgHom.coe_toRingHom]
· intro r
rw [hx.mvPolynomialOptionEquivPolynomialAdjoin_C, aeval_C, Polynomial.aeval_C,
IsScalarTower.algebraMap_apply R (adjoin R (range x)) A]
· rintro (⟨⟩ | ⟨i⟩)
· rw [hx.mvPolynomialOptionEquivPolynomialAdjoin_X_none, aeval_X, Polynomial.aeval_X,
Option.elim]
· rw [hx.mvPolynomialOptionEquivPolynomialAdjoin_X_some, Polynomial.aeval_C,
hx.algebraMap_aevalEquiv, aeval_X, aeval_X, Option.elim] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.Algebra.MvPolynomial.Monad",
"Mathlib.Algebra.MvPolynomial.Supported",
"Mathlib.RingTheory.AlgebraicIndependent.Defs",
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.RingTheory.MvPolynomial.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Basic.lean | AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin | null |
algebraicIndependent_empty_type [IsEmpty ι] [Nontrivial A] : AlgebraicIndependent K x := by
rw [algebraicIndependent_empty_type_iff]
exact RingHom.injective _ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.Algebra.MvPolynomial.Monad",
"Mathlib.Algebra.MvPolynomial.Supported",
"Mathlib.RingTheory.AlgebraicIndependent.Defs",
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.RingTheory.MvPolynomial.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Basic.lean | algebraicIndependent_empty_type | null |
algebraicIndependent_empty [Nontrivial A] :
AlgebraicIndependent K ((↑) : (∅ : Set A) → A) :=
algebraicIndependent_empty_type | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.Algebra.MvPolynomial.Equiv",
"Mathlib.Algebra.MvPolynomial.Monad",
"Mathlib.Algebra.MvPolynomial.Supported",
"Mathlib.RingTheory.AlgebraicIndependent.Defs",
"Mathlib.RingTheory.Ideal.Maps",
"Mathlib.RingTheory.MvPolynomial.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Basic.lean | algebraicIndependent_empty | null |
@[stacks 030E "(1)"] AlgebraicIndependent : Prop :=
Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A) | def | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | AlgebraicIndependent | `AlgebraicIndependent R x` states the family of elements `x`
is algebraically independent over `R`, meaning that the canonical
map out of the multivariable polynomial ring is injective. |
AlgebraicIndepOn (s : Set ι) : Prop := AlgebraicIndependent R fun i : s ↦ x i
variable {R} {x} | abbrev | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | AlgebraicIndepOn | `AlgebraicIndepOn R v s` states that the elements in the family `v` that are indexed by the
elements of `s` are algebraically independent over `R`. |
algebraicIndependent_iff :
AlgebraicIndependent R x ↔
∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=
injective_iff_map_eq_zero _ | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | algebraicIndependent_iff | null |
AlgebraicIndependent.eq_zero_of_aeval_eq_zero (h : AlgebraicIndependent R x) :
∀ p : MvPolynomial ι R, MvPolynomial.aeval (x : ι → A) p = 0 → p = 0 :=
algebraicIndependent_iff.1 h | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | AlgebraicIndependent.eq_zero_of_aeval_eq_zero | null |
algebraicIndependent_iff_injective_aeval :
AlgebraicIndependent R x ↔ Injective (MvPolynomial.aeval x : MvPolynomial ι R →ₐ[R] A) :=
Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | algebraicIndependent_iff_injective_aeval | null |
of_comp (f : A →ₐ[R] A') (hfv : AlgebraicIndependent R (f ∘ x)) :
AlgebraicIndependent R x := by
have : aeval (f ∘ x) = f.comp (aeval x) := by ext; simp
rw [AlgebraicIndependent, this, AlgHom.coe_comp] at hfv
exact hfv.of_comp
variable (hx : AlgebraicIndependent R x)
include hx | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | of_comp | null |
comp (f : ι' → ι) (hf : Function.Injective f) : AlgebraicIndependent R (x ∘ f) := by
intro p q
simpa [aeval_rename, (rename_injective f hf).eq_iff] using @hx (rename f p) (rename f q) | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | comp | null |
coe_range : AlgebraicIndependent R ((↑) : range x → A) := by
simpa using hx.comp _ (rangeSplitting_injective x) | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | coe_range | null |
algebraicIndependent_equiv (e : ι ≃ ι') {f : ι' → A} :
AlgebraicIndependent R (f ∘ e) ↔ AlgebraicIndependent R f :=
⟨fun h => Function.comp_id f ▸ e.self_comp_symm ▸ h.comp _ e.symm.injective,
fun h => h.comp _ e.injective⟩ | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | algebraicIndependent_equiv | null |
algebraicIndependent_equiv' (e : ι ≃ ι') {f : ι' → A} {g : ι → A} (h : f ∘ e = g) :
AlgebraicIndependent R g ↔ AlgebraicIndependent R f :=
h ▸ algebraicIndependent_equiv e | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | algebraicIndependent_equiv' | null |
algebraicIndependent_subtype_range {ι} {f : ι → A} (hf : Injective f) :
AlgebraicIndependent R ((↑) : range f → A) ↔ AlgebraicIndependent R f :=
Iff.symm <| algebraicIndependent_equiv' (Equiv.ofInjective f hf) rfl
alias ⟨AlgebraicIndependent.of_subtype_range, _⟩ := algebraicIndependent_subtype_range | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | algebraicIndependent_subtype_range | null |
algebraicIndependent_image {ι} {s : Set ι} {f : ι → A} (hf : Set.InjOn f s) :
(AlgebraicIndependent R fun x : s => f x) ↔ AlgebraicIndependent R fun x : f '' s => (x : A) :=
algebraicIndependent_equiv' (Equiv.Set.imageOfInjOn _ _ hf) rfl | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | algebraicIndependent_image | null |
mono {t s : Set A} (h : t ⊆ s)
(hx : AlgebraicIndependent R ((↑) : s → A)) : AlgebraicIndependent R ((↑) : t → A) := by
simpa [Function.comp] using hx.comp (inclusion h) (inclusion_injective h) | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | mono | null |
@[simps! apply_coe]
aevalEquiv : MvPolynomial ι R ≃ₐ[R] Algebra.adjoin R (range x) :=
(AlgEquiv.ofInjective (aeval x) (algebraicIndependent_iff_injective_aeval.1 hx)).trans
(Subalgebra.equivOfEq _ _ (Algebra.adjoin_range_eq_range_aeval R x).symm)
@[simp] | def | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | aevalEquiv | Canonical isomorphism between polynomials and the subalgebra generated by
algebraically independent elements. |
algebraMap_aevalEquiv (p : MvPolynomial ι R) :
algebraMap (Algebra.adjoin R (range x)) A (hx.aevalEquiv p) = aeval x p :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | algebraMap_aevalEquiv | null |
repr : Algebra.adjoin R (range x) →ₐ[R] MvPolynomial ι R :=
hx.aevalEquiv.symm
@[simp] | def | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | repr | The canonical map from the subalgebra generated by an algebraic independent family
into the polynomial ring. |
aeval_repr (p) : aeval x (hx.repr p) = p :=
Subtype.ext_iff.1 (AlgEquiv.apply_symm_apply hx.aevalEquiv p) | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | aeval_repr | null |
aeval_comp_repr : (aeval x).comp hx.repr = Subalgebra.val _ :=
AlgHom.ext hx.aeval_repr | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | aeval_comp_repr | null |
@[stacks 030E "(4)"] IsTranscendenceBasis (x : ι → A) : Prop :=
AlgebraicIndependent R x ∧
∀ (s : Set A) (_ : AlgebraicIndepOn R id s) (_ : range x ⊆ s), range x = s | def | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | IsTranscendenceBasis | A family is a transcendence basis if it is a maximal algebraically independent subset. |
isTranscendenceBasis_iff_maximal {s : Set A} :
IsTranscendenceBasis R ((↑) : s → A) ↔ Maximal (AlgebraicIndepOn R id) s := by
rw [IsTranscendenceBasis, maximal_iff, Subtype.range_val]; rfl | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | isTranscendenceBasis_iff_maximal | null |
isTranscendenceBasis_equiv (e : ι ≃ ι') {f : ι' → A} :
IsTranscendenceBasis R (f ∘ e) ↔ IsTranscendenceBasis R f := by
simp_rw [IsTranscendenceBasis, algebraicIndependent_equiv, EquivLike.range_comp]
alias ⟨_, IsTranscendenceBasis.comp_equiv⟩ := isTranscendenceBasis_equiv | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | isTranscendenceBasis_equiv | null |
isTranscendenceBasis_equiv' (e : ι ≃ ι') {f : ι' → A} {g : ι → A} (h : f ∘ e = g) :
IsTranscendenceBasis R g ↔ IsTranscendenceBasis R f :=
h ▸ isTranscendenceBasis_equiv e | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | isTranscendenceBasis_equiv' | null |
isTranscendenceBasis_subtype_range {ι} {f : ι → A} (hf : Injective f) :
IsTranscendenceBasis R ((↑) : range f → A) ↔ IsTranscendenceBasis R f :=
.symm <| isTranscendenceBasis_equiv' (Equiv.ofInjective f hf) rfl
alias ⟨IsTranscendenceBasis.of_subtype_range, _⟩ := isTranscendenceBasis_subtype_range | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | isTranscendenceBasis_subtype_range | null |
isTranscendenceBasis_image {ι} {s : Set ι} {f : ι → A} (hf : Set.InjOn f s) :
IsTranscendenceBasis R (fun x : s ↦ f x) ↔ IsTranscendenceBasis R fun x : f '' s ↦ (x : A) :=
isTranscendenceBasis_equiv' (Equiv.Set.imageOfInjOn _ _ hf) rfl | theorem | RingTheory | [
"Mathlib.Algebra.MvPolynomial.CommRing"
] | Mathlib/RingTheory/AlgebraicIndependent/Defs.lean | isTranscendenceBasis_image | null |
IsTranscendenceBasis.lift_cardinalMk_eq_max_lift
{F : Type u} {E : Type v} [CommRing F] [Nontrivial F] [CommRing E] [IsDomain E] [Algebra F E]
{ι : Type w} {x : ι → E} [Nonempty ι] (hx : IsTranscendenceBasis F x) :
lift.{max u w} #E = lift.{max v w} #F ⊔ lift.{max u v} #ι ⊔ ℵ₀ := by
let K := Algebra.adjoin F (Set.range x)
suffices #E = #K by simp [K, this, ← lift_mk_eq'.2 ⟨hx.1.aevalEquiv.toEquiv⟩]
haveI : Algebra.IsAlgebraic K E := hx.isAlgebraic
refine le_antisymm ?_ (mk_le_of_injective Subtype.val_injective)
haveI : Infinite K := hx.1.aevalEquiv.infinite_iff.1 inferInstance
simpa only [sup_eq_left.2 (aleph0_le_mk K)] using Algebra.IsAlgebraic.cardinalMk_le_max K E | theorem | RingTheory | [
"Mathlib.FieldTheory.IntermediateField.Adjoin.Basic",
"Mathlib.FieldTheory.MvRatFunc.Rank",
"Mathlib.RingTheory.Algebraic.Cardinality",
"Mathlib.RingTheory.AlgebraicIndependent.Adjoin",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental",
"Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis"
] | Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.lean | IsTranscendenceBasis.lift_cardinalMk_eq_max_lift | null |
IsTranscendenceBasis.lift_rank_eq_max_lift
{F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E]
{ι : Type w} {x : ι → E} [Nonempty ι] (hx : IsTranscendenceBasis F x) :
lift.{max u w} (Module.rank F E) = lift.{max v w} #F ⊔ lift.{max u v} #ι ⊔ ℵ₀ := by
let K := IntermediateField.adjoin F (Set.range x)
haveI : Algebra.IsAlgebraic K E := hx.isAlgebraic_field
rw [← rank_mul_rank F K E, lift_mul, ← hx.1.aevalEquivField.toLinearEquiv.lift_rank_eq,
MvRatFunc.rank_eq_max_lift, lift_max, lift_max, lift_lift, lift_lift, lift_aleph0]
refine mul_eq_left le_sup_right ((lift_le.2 ((rank_le_card K E).trans
(Algebra.IsAlgebraic.cardinalMk_le_max K E))).trans_eq ?_) (by simp [rank_pos.ne'])
simp [K, ← lift_mk_eq'.2 ⟨hx.1.aevalEquivField.toEquiv⟩] | theorem | RingTheory | [
"Mathlib.FieldTheory.IntermediateField.Adjoin.Basic",
"Mathlib.FieldTheory.MvRatFunc.Rank",
"Mathlib.RingTheory.Algebraic.Cardinality",
"Mathlib.RingTheory.AlgebraicIndependent.Adjoin",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental",
"Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis"
] | Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.lean | IsTranscendenceBasis.lift_rank_eq_max_lift | null |
Algebra.Transcendental.rank_eq_cardinalMk
(F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [Algebra.Transcendental F E] :
Module.rank F E = #E := by
obtain ⟨ι, x, hx⟩ := exists_isTranscendenceBasis' F E
haveI := hx.nonempty_iff_transcendental.2 ‹_›
simpa [← hx.lift_cardinalMk_eq_max_lift] using hx.lift_rank_eq_max_lift | theorem | RingTheory | [
"Mathlib.FieldTheory.IntermediateField.Adjoin.Basic",
"Mathlib.FieldTheory.MvRatFunc.Rank",
"Mathlib.RingTheory.Algebraic.Cardinality",
"Mathlib.RingTheory.AlgebraicIndependent.Adjoin",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental",
"Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis"
] | Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.lean | Algebra.Transcendental.rank_eq_cardinalMk | null |
IntermediateField.rank_sup_le
{F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (A B : IntermediateField F E) :
Module.rank F ↥(A ⊔ B) ≤ Module.rank F A * Module.rank F B := by
by_cases hA : Algebra.IsAlgebraic F A
· exact rank_sup_le_of_isAlgebraic A B (Or.inl hA)
by_cases hB : Algebra.IsAlgebraic F B
· exact rank_sup_le_of_isAlgebraic A B (Or.inr hB)
rw [← Algebra.transcendental_iff_not_isAlgebraic] at hA hB
haveI : Algebra.Transcendental F ↥(A ⊔ B) := .ringHom_of_comp_eq (RingHom.id F)
(inclusion le_sup_left) Function.surjective_id (inclusion_injective _) rfl
haveI := Algebra.Transcendental.infinite F A
haveI := Algebra.Transcendental.infinite F B
simp_rw [Algebra.Transcendental.rank_eq_cardinalMk]
rw [sup_def, mul_mk_eq_max, ← Cardinal.lift_le.{u}]
refine (lift_cardinalMk_adjoin_le _ _).trans ?_
calc
_ ≤ Cardinal.lift.{v} #F ⊔ Cardinal.lift.{u} (#A ⊔ #B) ⊔ ℵ₀ := by
gcongr
rw [Cardinal.lift_le]
exact (mk_union_le _ _).trans_eq (by simp)
_ = _ := by
simp [lift_mk_le_lift_mk_of_injective (algebraMap F A).injective] | theorem | RingTheory | [
"Mathlib.FieldTheory.IntermediateField.Adjoin.Basic",
"Mathlib.FieldTheory.MvRatFunc.Rank",
"Mathlib.RingTheory.Algebraic.Cardinality",
"Mathlib.RingTheory.AlgebraicIndependent.Adjoin",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental",
"Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis"
] | Mathlib/RingTheory/AlgebraicIndependent/RankAndCardinality.lean | IntermediateField.rank_sup_le | null |
exists_isTranscendenceBasis_superset {s : Set A}
(hs : AlgebraicIndepOn R id s) :
∃ t, s ⊆ t ∧ IsTranscendenceBasis R ((↑) : t → A) := by
simpa [← isTranscendenceBasis_iff_maximal]
using exists_maximal_algebraicIndependent s _ (subset_univ _) hs
variable (A) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | exists_isTranscendenceBasis_superset | null |
exists_isTranscendenceBasis [FaithfulSMul R A] :
∃ s : Set A, IsTranscendenceBasis R ((↑) : s → A) := by
simpa using exists_isTranscendenceBasis_superset
((algebraicIndependent_empty_iff R A).mpr (FaithfulSMul.algebraMap_injective R A)) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | exists_isTranscendenceBasis | null |
exists_isTranscendenceBasis' [FaithfulSMul R A] :
∃ (ι : Type w) (x : ι → A), IsTranscendenceBasis R x :=
have ⟨s, h⟩ := exists_isTranscendenceBasis R A
⟨s, Subtype.val, h⟩
variable {A}
open Cardinal in | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | exists_isTranscendenceBasis' | `Type` version of `exists_isTranscendenceBasis`. |
trdeg_eq_iSup_cardinalMk_isTranscendenceBasis :
trdeg R A = ⨆ ι : { s : Set A // IsTranscendenceBasis R ((↑) : s → A) }, #ι.1 := by
refine (ciSup_le' fun s ↦ ?_).antisymm
(ciSup_le' fun s ↦ le_ciSup_of_le (bddAbove_range _) ⟨s, s.2.1⟩ le_rfl)
choose t ht using exists_isTranscendenceBasis_superset s.2
exact le_ciSup_of_le (bddAbove_range _) ⟨t, ht.2⟩ (mk_le_mk_of_subset ht.1)
variable {R} | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | trdeg_eq_iSup_cardinalMk_isTranscendenceBasis | null |
AlgebraicIndependent.isTranscendenceBasis_iff [Nontrivial R]
(i : AlgebraicIndependent R x) :
IsTranscendenceBasis R x ↔
∀ (κ : Type w) (w : κ → A) (_ : AlgebraicIndependent R w) (j : ι → κ) (_ : w ∘ j = x),
Surjective j := by
fconstructor
· rintro p κ w i' j rfl
have p := p.2 (range w) i'.coe_range (range_comp_subset_range _ _)
rw [range_comp, ← @image_univ _ _ w] at p
exact range_eq_univ.mp (image_injective.mpr i'.injective p)
· intro p
use i
intro w i' h
specialize p w ((↑) : w → A) i' (fun i => ⟨x i, range_subset_iff.mp h i⟩) (by ext; simp)
have q := congr_arg (fun s => ((↑) : w → A) '' s) p.range_eq
dsimp at q
rw [← image_univ, image_image] at q
simpa using q | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | AlgebraicIndependent.isTranscendenceBasis_iff | null |
IsTranscendenceBasis.isAlgebraic [Nontrivial R] (hx : IsTranscendenceBasis R x) :
Algebra.IsAlgebraic (adjoin R (range x)) A := by
constructor
intro a
rw [← not_iff_comm.1 (hx.1.option_iff_transcendental _).symm]
intro ai
have h₁ : range x ⊆ range fun o : Option ι => o.elim a x := by
rintro x ⟨y, rfl⟩
exact ⟨some y, rfl⟩
have h₂ : range x ≠ range fun o : Option ι => o.elim a x := by
intro h
have : a ∈ range x := by
rw [h]
exact ⟨none, rfl⟩
rcases this with ⟨b, rfl⟩
have : some b = none := ai.injective rfl
simpa
exact h₂ (hx.2 (Set.range fun o : Option ι => o.elim a x)
((algebraicIndependent_subtype_range ai.injective).2 ai) h₁) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.isAlgebraic | null |
AlgebraicIndependent.isTranscendenceBasis_iff_isAlgebraic
[Nontrivial R] (ind : AlgebraicIndependent R x) :
IsTranscendenceBasis R x ↔ Algebra.IsAlgebraic (adjoin R (range x)) A := by
refine ⟨(·.isAlgebraic), fun alg ↦ ⟨ind, fun s ind_s hxs ↦ of_not_not fun hxs' ↦ ?_⟩⟩
have : ¬ s ⊆ range x := (hxs' <| hxs.antisymm ·)
have ⟨a, has, hax⟩ := not_subset.mp this
rw [show range x = Subtype.val '' range (Set.inclusion hxs) by
rw [← range_comp, val_comp_inclusion, Subtype.range_val]] at alg
refine ind_s.transcendental_adjoin (s := range (inclusion hxs)) (i := ⟨a, has⟩) ?_ (alg.1 _)
simpa using hax | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | AlgebraicIndependent.isTranscendenceBasis_iff_isAlgebraic | null |
isTranscendenceBasis_iff_algebraicIndependent_isAlgebraic [Nontrivial R] :
IsTranscendenceBasis R x ↔
AlgebraicIndependent R x ∧ Algebra.IsAlgebraic (adjoin R (range x)) A :=
⟨fun h ↦ ⟨h.1, h.1.isTranscendenceBasis_iff_isAlgebraic.mp h⟩,
fun ⟨ind, alg⟩ ↦ ind.isTranscendenceBasis_iff_isAlgebraic.mpr alg⟩ | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isTranscendenceBasis_iff_algebraicIndependent_isAlgebraic | null |
IsTranscendenceBasis.algebraMap_comp
[Nontrivial R] [NoZeroDivisors S] [Algebra.IsAlgebraic S A] [FaithfulSMul S A]
{x : ι → S} (hx : IsTranscendenceBasis R x) : IsTranscendenceBasis R (algebraMap S A ∘ x) := by
let f := IsScalarTower.toAlgHom R S A
refine hx.1.map (f := f) (FaithfulSMul.algebraMap_injective S A).injOn
|>.isTranscendenceBasis_iff_isAlgebraic.mpr ?_
rw [Set.range_comp, ← AlgHom.map_adjoin]
set Rx := adjoin R (range x)
let e := Rx.equivMapOfInjective f (FaithfulSMul.algebraMap_injective S A)
letI := e.toRingHom.toAlgebra
haveI : IsScalarTower Rx (Rx.map f) A := .of_algebraMap_eq fun x ↦ rfl
have : Algebra.IsAlgebraic Rx S := hx.isAlgebraic
have : Algebra.IsAlgebraic Rx A := .trans _ S _
exact .extendScalars e.injective | lemma | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.algebraMap_comp | null |
IsTranscendenceBasis.isAlgebraic_iff [IsDomain S] [NoZeroDivisors A]
{ι : Type*} {v : ι → A} (hv : IsTranscendenceBasis R v) :
Algebra.IsAlgebraic S A ↔ ∀ i, IsAlgebraic S (v i) := by
refine ⟨fun _ i ↦ Algebra.IsAlgebraic.isAlgebraic (v i), fun H ↦ ?_⟩
let Rv := adjoin R (range v)
let Sv := adjoin S (range v)
have : Algebra.IsAlgebraic S Sv := by
simpa [Sv, ← Subalgebra.isAlgebraic_iff, isAlgebraic_adjoin_iff]
have le : Rv ≤ Sv.restrictScalars R := by
rw [Subalgebra.restrictScalars_adjoin]; exact le_sup_right
letI : Algebra Rv Sv := (Subalgebra.inclusion le).toAlgebra
have : IsScalarTower Rv Sv A := .of_algebraMap_eq fun x ↦ rfl
have := (algebraMap R S).domain_nontrivial
have := hv.isAlgebraic
have : Algebra.IsAlgebraic Sv A := .extendScalars (Subalgebra.inclusion_injective le)
exact .trans _ Sv _
variable (ι R) | lemma | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.isAlgebraic_iff | null |
IsTranscendenceBasis.mvPolynomial [Nontrivial R] :
IsTranscendenceBasis R (X (R := R) (σ := ι)) := by
refine isTranscendenceBasis_iff_algebraicIndependent_isAlgebraic.2 ⟨algebraicIndependent_X .., ?_⟩
rw [adjoin_range_X]
set A := MvPolynomial ι R
have := Algebra.isIntegral_of_surjective (R := (⊤ : Subalgebra R A)) (B := A) (⟨⟨·, ⟨⟩⟩, rfl⟩)
infer_instance | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.mvPolynomial | null |
IsTranscendenceBasis.mvPolynomial' [Nonempty ι] :
IsTranscendenceBasis R (X (R := R) (σ := ι)) := by nontriviality R; exact .mvPolynomial ι R | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.mvPolynomial' | null |
IsTranscendenceBasis.polynomial [Nonempty ι] [Subsingleton ι] :
IsTranscendenceBasis R fun _ : ι ↦ (.X : Polynomial R) := by
nontriviality R
have := (nonempty_unique ι).some
refine (isTranscendenceBasis_equiv (Equiv.equivPUnit.{_, 1} _).symm).mp <|
(MvPolynomial.pUnitAlgEquiv R).symm.isTranscendenceBasis_iff.mp ?_
convert IsTranscendenceBasis.mvPolynomial PUnit R
ext; simp
variable {ι R} | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.polynomial | null |
IsTranscendenceBasis.sumElim_comp [NoZeroDivisors A] {x : ι → S} {y : ι' → A}
(hx : IsTranscendenceBasis R x) (hy : IsTranscendenceBasis S y) :
IsTranscendenceBasis R (Sum.elim y (algebraMap S A ∘ x)) := by
cases subsingleton_or_nontrivial R
· rw [isTranscendenceBasis_iff_of_subsingleton] at hx ⊢; infer_instance
rw [(hx.1.sumElim_comp hy.1).isTranscendenceBasis_iff_isAlgebraic]
set Rx := adjoin R (range x)
let Rxy := adjoin Rx (range y)
rw [show adjoin R (range <| Sum.elim y (algebraMap S A ∘ x)) = Rxy.restrictScalars R by
rw [← adjoin_algebraMap_image_union_eq_adjoin_adjoin, Sum.elim_range, union_comm, range_comp]]
change Algebra.IsAlgebraic Rxy A
have := hx.1.algebraMap_injective.nontrivial
have := hy.1.algebraMap_injective.nontrivial
have := hy.isAlgebraic
set Sy := adjoin S (range y)
let _ : Algebra Rxy Sy := by
refine (Subalgebra.inclusion (T := Sy.restrictScalars Rx) <| adjoin_le ?_).toAlgebra
rintro _ ⟨i, rfl⟩; exact subset_adjoin (s := range y) ⟨i, rfl⟩
have : IsScalarTower Rxy Sy A := .of_algebraMap_eq fun ⟨a, _⟩ ↦ show a = _ from rfl
have : IsScalarTower Rx Rxy Sy := .of_algebraMap_eq fun ⟨a, _⟩ ↦ Subtype.ext rfl
have : Algebra.IsAlgebraic Rxy Sy := by
refine ⟨fun ⟨a, ha⟩ ↦ adjoin_induction ?_ (fun _ ↦ .extendScalars (R := Rx) ?_ ?_)
(fun _ _ _ _ ↦ .add) (fun _ _ _ _ ↦ .mul) ha⟩
· rintro _ ⟨i, rfl⟩; exact isAlgebraic_algebraMap (⟨y i, subset_adjoin ⟨i, rfl⟩⟩ : Rxy)
· exact fun _ _ ↦ (Subtype.ext <| hy.1.algebraMap_injective <| Subtype.ext_iff.mp ·)
· exact (hx.isAlgebraic.1 _).algHom (IsScalarTower.toAlgHom Rx S Sy)
exact .trans _ Sy _ | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.sumElim_comp | null |
IsTranscendenceBasis.isEmpty_iff_isAlgebraic [Nontrivial R]
(hx : IsTranscendenceBasis R x) :
IsEmpty ι ↔ Algebra.IsAlgebraic R A := by
refine ⟨fun _ ↦ ?_, fun _ ↦ hx.1.isEmpty_of_isAlgebraic⟩
have := hx.isAlgebraic
rw [Set.range_eq_empty x, adjoin_empty] at this
exact algebra_isAlgebraic_of_algebra_isAlgebraic_bot_left R A | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.isEmpty_iff_isAlgebraic | If `x` is a transcendence basis of `A/R`, then it is empty if and only if
`A/R` is algebraic. |
IsTranscendenceBasis.nonempty_iff_transcendental [Nontrivial R]
(hx : IsTranscendenceBasis R x) :
Nonempty ι ↔ Algebra.Transcendental R A := by
rw [← not_isEmpty_iff, Algebra.transcendental_iff_not_isAlgebraic, hx.isEmpty_iff_isAlgebraic] | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.nonempty_iff_transcendental | If `x` is a transcendence basis of `A/R`, then it is not empty if and only if
`A/R` is transcendental. |
IsTranscendenceBasis.isAlgebraic_field {F E : Type*} {x : ι → E}
[Field F] [Field E] [Algebra F E] (hx : IsTranscendenceBasis F x) :
Algebra.IsAlgebraic (IntermediateField.adjoin F (range x)) E := by
haveI := hx.isAlgebraic
set S := range x
letI : Algebra (adjoin F S) (IntermediateField.adjoin F S) :=
(Subalgebra.inclusion (IntermediateField.algebra_adjoin_le_adjoin F S)).toRingHom.toAlgebra
haveI : IsScalarTower (adjoin F S) (IntermediateField.adjoin F S) E :=
IsScalarTower.of_algebraMap_eq (congrFun rfl)
exact Algebra.IsAlgebraic.extendScalars (R := adjoin F S) (Subalgebra.inclusion_injective _) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | IsTranscendenceBasis.isAlgebraic_field | null |
private indepMatroid : IndepMatroid A where
E := univ
Indep := AlgebraicIndepOn R id
indep_empty := (algebraicIndependent_empty_iff ..).mpr (FaithfulSMul.algebraMap_injective R A)
indep_subset _ _ := (·.mono)
indep_aug I B I_ind h B_base := by
contrapose! h
rw [← isTranscendenceBasis_iff_maximal] at B_base ⊢
cases subsingleton_or_nontrivial R
· rw [isTranscendenceBasis_iff_of_subsingleton] at B_base ⊢
by_contra this
have ⟨b, hb⟩ := B_base
exact h b ⟨hb, fun hbI ↦ this ⟨b, hbI⟩⟩ .of_subsingleton
apply I_ind.isTranscendenceBasis_iff_isAlgebraic.mpr
replace B_base := B_base.isAlgebraic
simp_rw [id_eq]
rw [Subtype.range_val] at B_base ⊢
refine ⟨fun a ↦ (B_base.1 a).adjoin_of_forall_isAlgebraic fun x hx ↦ ?_⟩
contrapose! h
exact ⟨x, hx, I_ind.insert <| by rwa [image_id]⟩
indep_maximal X _ I ind hIX := exists_maximal_algebraicIndependent I X hIX ind
subset_ground _ _ := subset_univ _ | def | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | indepMatroid | null |
matroid : Matroid A := (indepMatroid R A).matroid.copyBase univ
(fun s ↦ IsTranscendenceBasis R ((↑) : s → A)) rfl
(fun B ↦ by simp_rw [Matroid.isBase_iff_maximal_indep, isTranscendenceBasis_iff_maximal]; rfl) | def | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid | If `R` is a commutative ring and `A` is a commutative `R`-algebra with injective algebra map
and no zero-divisors, then the `R`-algebraic independent subsets of `A` form a matroid. |
@[simp] matroid_e : (matroid R A).E = univ := rfl | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_e | null |
matroid_cRank_eq : (matroid R A).cRank = trdeg R A :=
(trdeg_eq_iSup_cardinalMk_isTranscendenceBasis _).symm
variable {R A} | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_cRank_eq | null |
matroid_indep_iff {s : Set A} :
(matroid R A).Indep s ↔ AlgebraicIndepOn R id s := Iff.rfl | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_indep_iff | null |
matroid_isBase_iff {s : Set A} :
(matroid R A).IsBase s ↔ IsTranscendenceBasis R ((↑) : s → A) := Iff.rfl | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_isBase_iff | null |
matroid_isBasis_iff [IsDomain A] {s t : Set A} : (matroid R A).IsBasis s t ↔
AlgebraicIndepOn R id s ∧ s ⊆ t ∧ ∀ a ∈ t, IsAlgebraic (adjoin R s) a := by
rw [Matroid.IsBasis, maximal_iff_forall_insert fun s t h hst ↦ ⟨h.1.subset hst, hst.trans h.2⟩]
simp_rw [matroid_indep_iff, ← and_assoc, matroid_e, subset_univ, and_true]
exact and_congr_right fun h ↦ ⟨fun max a ha ↦ of_not_not fun tr ↦ max _
(fun ha ↦ tr (isAlgebraic_algebraMap (⟨a, subset_adjoin ha⟩ : adjoin R s)))
⟨.insert h.1 (by rwa [image_id]), insert_subset ha h.2⟩,
fun alg a ha h ↦ ((AlgebraicIndepOn.insert_iff ha).mp h.1).2 <| by
rw [image_id]; exact alg _ <| h.2 <| mem_insert ..⟩
open Subsingleton in | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_isBasis_iff | null |
matroid_isBasis_iff_of_subsingleton [Subsingleton A] {s t : Set A} :
(matroid R A).IsBasis s t ↔ s = t := by
have := (FaithfulSMul.algebraMap_injective R A).subsingleton
simp_rw [Matroid.IsBasis, matroid_indep_iff, of_subsingleton, true_and,
matroid_e, subset_univ, and_true, ← le_iff_subset, maximal_le_iff] | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_isBasis_iff_of_subsingleton | null |
isAlgebraic_adjoin_iff_of_matroid_isBasis [NoZeroDivisors A] {s t : Set A} {a : A}
(h : (matroid R A).IsBasis s t) : IsAlgebraic (adjoin R s) a ↔ IsAlgebraic (adjoin R t) a := by
cases subsingleton_or_nontrivial A
· apply iff_of_false <;> apply is_transcendental_of_subsingleton
have := (isDomain_iff_noZeroDivisors_and_nontrivial A).mpr ⟨inferInstance, inferInstance⟩
exact ⟨(·.adjoin_of_forall_isAlgebraic fun x hx ↦ (hx.2 <| h.1.1.2 hx.1).elim),
(·.adjoin_of_forall_isAlgebraic fun x hx ↦ (matroid_isBasis_iff.mp h).2.2 _ hx.1)⟩ | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isAlgebraic_adjoin_iff_of_matroid_isBasis | null |
matroid_closure_eq [IsDomain A] {s : Set A} :
(matroid R A).closure s = algebraicClosure (adjoin R s) A := by
have ⟨B, hB⟩ := (matroid R A).exists_isBasis s
simp_rw [← hB.closure_eq_closure, hB.1.1.1.closure_eq_setOf_isBasis_insert, Set.ext_iff,
mem_setOf, matroid_isBasis_iff, ← matroid_indep_iff, hB.1.1.1, subset_insert, true_and,
SetLike.mem_coe, mem_algebraicClosure, ← isAlgebraic_adjoin_iff_of_matroid_isBasis hB,
forall_mem_insert]
exact fun _ ↦ and_iff_left fun x hx ↦ isAlgebraic_algebraMap (⟨x, subset_adjoin hx⟩ : adjoin R B) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_closure_eq | null |
matroid_isFlat_iff [IsDomain A] {s : Set A} :
(matroid R A).IsFlat s ↔ ∃ S : Subalgebra R A, S = s ∧ ∀ a : A, IsAlgebraic S a → a ∈ s := by
rw [Matroid.isFlat_iff_closure_eq, matroid_closure_eq]
set S := algebraicClosure (adjoin R s) A
refine ⟨fun eq ↦ ⟨S.restrictScalars R, eq, fun a (h : IsAlgebraic S _) ↦ ?_⟩, ?_⟩
· rw [← eq]; exact h.restrictScalars (adjoin R s)
rintro ⟨s, rfl, hs⟩
refine Set.ext fun a ↦ ⟨(hs _ <| adjoin_eq s ▸ ·), fun h ↦ ?_⟩
exact isAlgebraic_algebraMap (A := A) (by exact (⟨a, subset_adjoin h⟩ : adjoin R s)) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_isFlat_iff | null |
matroid_spanning_iff [IsDomain A] {s : Set A} :
(matroid R A).Spanning s ↔ Algebra.IsAlgebraic (adjoin R s) A := by
simp_rw [Matroid.spanning_iff, matroid_e, subset_univ, and_true, eq_univ_iff_forall,
matroid_closure_eq, SetLike.mem_coe, mem_algebraicClosure, Algebra.isAlgebraic_def]
open Subsingleton -- brings the Subsingleton.to_noZeroDivisors instance into scope | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_spanning_iff | null |
matroid_isFlat_of_subsingleton [Subsingleton A] (s : Set A) : (matroid R A).IsFlat s := by
simp_rw [Matroid.isFlat_iff, matroid_e, subset_univ,
and_true, matroid_isBasis_iff_of_subsingleton]
exact fun I X hIs hIX ↦ (hIX.symm.trans hIs).subset | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_isFlat_of_subsingleton | null |
matroid_closure_of_subsingleton [Subsingleton A] (s : Set A) :
(matroid R A).closure s = s := by
simp_rw [Matroid.closure, matroid_isFlat_of_subsingleton, true_and, matroid_e, inter_univ]
exact subset_antisymm (sInter_subset_of_mem <| subset_refl s) (subset_sInter fun _ ↦ id) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_closure_of_subsingleton | null |
matroid_spanning_iff_of_subsingleton [Subsingleton A] {s : Set A} :
(matroid R A).Spanning s ↔ s = univ := by
simp_rw [Matroid.spanning_iff, matroid_closure_of_subsingleton, matroid_e, subset_univ, and_true] | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | matroid_spanning_iff_of_subsingleton | null |
exists_isTranscendenceBasis_between [NoZeroDivisors A] (s t : Set A) (hst : s ⊆ t)
(hs : AlgebraicIndepOn R id s) [ht : Algebra.IsAlgebraic (adjoin R t) A] :
∃ u, s ⊆ u ∧ u ⊆ t ∧ IsTranscendenceBasis R ((↑) : u → A) := by
have := ht.nontrivial
have := Subtype.val_injective (p := (· ∈ adjoin R t)).nontrivial
have := (isDomain_iff_noZeroDivisors_and_nontrivial A).mpr ⟨inferInstance, inferInstance⟩
have := (faithfulSMul_iff_algebraMap_injective R A).mpr hs.algebraMap_injective
rw [← matroid_spanning_iff] at ht
rw [← matroid_indep_iff] at hs
have ⟨B, base, hsB, hBt⟩ := hs.exists_isBase_subset_spanning ht hst
exact ⟨B, hsB, hBt, base⟩ | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | exists_isTranscendenceBasis_between | If `s ⊆ t` are subsets in an `R`-algebra `A` such that `s` is algebraically independent over
`R`, and `A` is algebraic over the `R`-algebra generated by `t`, then there is a transcendence
basis of `A` over `R` between `s` and `t`, provided that `A` is a domain.
This may fail if only `R` is assumed to be a domain but `A` is not, because of failure of
transitivity of algebraicity: there may exist `a : A` such that `S := R[a]` is algebraic over
`R` and `A` is algebraic over `S`, but `A` nonetheless contains a transcendental element over `R`.
The only `R`-algebraically independent subset of `{a}` is `∅`, which is not a transcendence basis.
See the docstring of `IsAlgebraic.restrictScalars_of_isIntegral` for an example. |
exists_isTranscendenceBasis_subset [NoZeroDivisors A] [FaithfulSMul R A]
(s : Set A) [Algebra.IsAlgebraic (adjoin R s) A] :
∃ t, t ⊆ s ∧ IsTranscendenceBasis R ((↑) : t → A) := by
have ⟨t, _, ht⟩ := exists_isTranscendenceBasis_between ∅ s (empty_subset _)
((algebraicIndependent_empty_iff ..).mpr <| FaithfulSMul.algebraMap_injective R A)
exact ⟨t, ht⟩ | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | exists_isTranscendenceBasis_subset | null |
isAlgebraic_iff_exists_isTranscendenceBasis_subset
[IsDomain A] [FaithfulSMul R A] {s : Set A} :
Algebra.IsAlgebraic (adjoin R s) A ↔ ∃ t, t ⊆ s ∧ IsTranscendenceBasis R ((↑) : t → A) := by
simp_rw [← matroid_spanning_iff, ← matroid_isBase_iff, and_comm (a := _ ⊆ _)]
exact Matroid.spanning_iff_exists_isBase_subset (subset_univ _)
open Cardinal AlgebraicIndependent | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isAlgebraic_iff_exists_isTranscendenceBasis_subset | null |
lift_cardinalMk_eq_trdeg (hx : IsTranscendenceBasis R x) :
lift.{w} #ι = lift.{u} (trdeg R A) := by
have := (faithfulSMul_iff_algebraMap_injective R A).mpr hx.1.algebraMap_injective
rw [← matroid_cRank_eq, ← ((matroid_isBase_iff).mpr hx.to_subtype_range).cardinalMk_eq_cRank,
lift_mk_eq'.mpr ⟨.ofInjective _ hx.1.injective⟩] | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | lift_cardinalMk_eq_trdeg | null |
cardinalMk_eq_trdeg {ι : Type w} {x : ι → A} (hx : IsTranscendenceBasis R x) :
#ι = trdeg R A := by
rw [← lift_id #ι, lift_cardinalMk_eq_trdeg hx, lift_id] | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | cardinalMk_eq_trdeg | null |
@[stacks 030F]
lift_cardinalMk_eq (hx : IsTranscendenceBasis R x) (hy : IsTranscendenceBasis R y) :
lift.{u'} #ι = lift.{u} #ι' := by
rw [← lift_inj.{_, w}, lift_lift, lift_lift, ← lift_lift.{w, u'}, hx.lift_cardinalMk_eq_trdeg,
← lift_lift.{w, u}, hy.lift_cardinalMk_eq_trdeg, lift_lift, lift_lift]
@[stacks 030F] theorem cardinalMk_eq {ι' : Type u} {y : ι' → A}
(hx : IsTranscendenceBasis R x) (hy : IsTranscendenceBasis R y) :
#ι = #ι' := by
rw [← lift_id #ι, lift_cardinalMk_eq hx hy, lift_id] | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | lift_cardinalMk_eq | Any two transcendence bases of a domain `A` have the same cardinality.
May fail if `A` is not a domain; see https://mathoverflow.net/a/144580. |
@[simp]
MvPolynomial.trdeg_of_isDomain [IsDomain S] : trdeg S (MvPolynomial ι S) = lift.{v} #ι := by
have := (IsTranscendenceBasis.mvPolynomial ι S).lift_cardinalMk_eq_trdeg.symm
rwa [lift_id', ← lift_lift.{u}, lift_id] at this
@[simp] | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | MvPolynomial.trdeg_of_isDomain | null |
Polynomial.trdeg_of_isDomain [IsDomain R] : trdeg R (Polynomial R) = 1 := by
simpa using (IsTranscendenceBasis.polynomial Unit R).lift_cardinalMk_eq_trdeg.symm | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | Polynomial.trdeg_of_isDomain | null |
trdeg_lt_aleph0 [IsDomain R] [fin : FiniteType R S] : trdeg R S < ℵ₀ :=
have ⟨n, f, surj⟩ := FiniteType.iff_quotient_mvPolynomial''.mp fin
lift_lt.mp <| (lift_trdeg_le_of_surjective f surj).trans_lt <| by
simpa using Cardinal.nat_lt_aleph0 _ | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | trdeg_lt_aleph0 | null |
isDomain_of_adjoin_range [Algebra.IsAlgebraic (adjoin R s) A] : IsDomain A :=
have := Algebra.IsAlgebraic.nontrivial (adjoin R s) A
(isDomain_iff_noZeroDivisors_and_nontrivial _).mpr
⟨‹_›, (Subtype.val_injective (p := (· ∈ adjoin R s))).nontrivial⟩ | lemma | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isDomain_of_adjoin_range | null |
trdeg_le_cardinalMk [alg : Algebra.IsAlgebraic (adjoin R s) A] : trdeg R A ≤ #s := by
by_cases h : Injective (algebraMap R A)
on_goal 2 => simp [trdeg_eq_zero_of_not_injective h]
have := isDomain_of_adjoin_range R s
have := (faithfulSMul_iff_algebraMap_injective R A).mpr h
rw [← matroid_spanning_iff, ← matroid_cRank_eq] at *
exact alg.cRank_le_cardinalMk
variable [FaithfulSMul R A] | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | trdeg_le_cardinalMk | null |
isTranscendenceBasis_of_lift_le_trdeg_of_finite
[Finite ι] [alg : Algebra.IsAlgebraic (adjoin R (range x)) A]
(le : lift.{w} #ι ≤ lift.{u} (trdeg R A)) : IsTranscendenceBasis R x := by
have ⟨_, h⟩ := lift_mk_le'.mp (le.trans <| lift_le.mpr <| trdeg_le_cardinalMk R (range x))
have := rangeFactorization_surjective.bijective_of_nat_card_le (Nat.card_le_card_of_injective _ h)
refine .of_subtype_range (fun _ _ ↦ (this.1 <| Subtype.ext ·)) ?_
have := isDomain_of_adjoin_range R (range x)
rw [← matroid_spanning_iff, ← matroid_cRank_eq] at *
exact alg.isBase_of_le_cRank_of_finite (lift_le.mp <| mk_range_le_lift.trans le) (finite_range x) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isTranscendenceBasis_of_lift_le_trdeg_of_finite | null |
isTranscendenceBasis_of_le_trdeg_of_finite {ι : Type w} [Finite ι] (x : ι → A)
[Algebra.IsAlgebraic (adjoin R (range x)) A] (le : #ι ≤ trdeg R A) :
IsTranscendenceBasis R x :=
isTranscendenceBasis_of_lift_le_trdeg_of_finite R x (by rwa [lift_id, lift_id]) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isTranscendenceBasis_of_le_trdeg_of_finite | null |
isTranscendenceBasis_of_lift_le_trdeg [Algebra.IsAlgebraic (adjoin R (range x)) A]
(fin : trdeg R A < ℵ₀) (le : lift.{w} #ι ≤ lift.{u} (trdeg R A)) :
IsTranscendenceBasis R x :=
have := mk_lt_aleph0_iff.mp (lift_lt.mp <| le.trans_lt <| (lift_lt.mpr fin).trans_eq <| by simp)
isTranscendenceBasis_of_lift_le_trdeg_of_finite R x le | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isTranscendenceBasis_of_lift_le_trdeg | null |
isTranscendenceBasis_of_le_trdeg {ι : Type w} (x : ι → A)
[Algebra.IsAlgebraic (adjoin R (range x)) A] (fin : trdeg R A < ℵ₀)
(le : #ι ≤ trdeg R A) : IsTranscendenceBasis R x :=
isTranscendenceBasis_of_lift_le_trdeg R x fin (by rwa [lift_id, lift_id]) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isTranscendenceBasis_of_le_trdeg | null |
isTranscendenceBasis_of_lift_trdeg_le (hx : AlgebraicIndependent R x)
(fin : trdeg R A < ℵ₀) (le : lift.{u} (trdeg R A) ≤ lift.{w} #ι) :
IsTranscendenceBasis R x := by
have := (faithfulSMul_iff_algebraMap_injective R A).mpr hx.algebraMap_injective
rw [← matroid_cRank_eq, ← Matroid.rankFinite_iff_cRank_lt_aleph0] at fin
exact .of_subtype_range hx.injective <| matroid_indep_iff.mpr hx.to_subtype_range
|>.isBase_of_cRank_le <| lift_le.mp <| (matroid_cRank_eq R A ▸ le).trans_eq
(mk_range_eq_of_injective hx.injective).symm | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isTranscendenceBasis_of_lift_trdeg_le | null |
isTranscendenceBasis_of_trdeg_le {ι : Type w} {x : ι → A} (hx : AlgebraicIndependent R x)
(fin : trdeg R A < ℵ₀) (le : trdeg R A ≤ #ι) : IsTranscendenceBasis R x :=
isTranscendenceBasis_of_lift_trdeg_le hx fin (by rwa [lift_id, lift_id]) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isTranscendenceBasis_of_trdeg_le | null |
isTranscendenceBasis_of_lift_trdeg_le_of_finite [Finite ι] (hx : AlgebraicIndependent R x)
(le : lift.{u} (trdeg R A) ≤ lift.{w} #ι) : IsTranscendenceBasis R x :=
isTranscendenceBasis_of_lift_trdeg_le hx
(lift_lt.mp <| le.trans_lt <| by simp) le | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isTranscendenceBasis_of_lift_trdeg_le_of_finite | null |
isTranscendenceBasis_of_trdeg_le_of_finite {ι : Type w} [Finite ι] {x : ι → A}
(hx : AlgebraicIndependent R x) (le : trdeg R A ≤ #ι) : IsTranscendenceBasis R x :=
isTranscendenceBasis_of_lift_trdeg_le_of_finite hx (by rwa [lift_id, lift_id]) | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | isTranscendenceBasis_of_trdeg_le_of_finite | null |
@[stacks 030H] lift_trdeg_add_eq [Nontrivial R] [NoZeroDivisors A] [FaithfulSMul R S]
[FaithfulSMul S A] : lift.{w} (trdeg R S) + lift.{v} (trdeg S A) = lift.{v} (trdeg R A) := by
have ⟨s, hs⟩ := exists_isTranscendenceBasis R S
have ⟨t, ht⟩ := exists_isTranscendenceBasis S A
have := (FaithfulSMul.algebraMap_injective S A).noZeroDivisors _ (map_zero _) (map_mul _)
have := (FaithfulSMul.algebraMap_injective R S).nontrivial
rw [← hs.cardinalMk_eq_trdeg, ← ht.cardinalMk_eq_trdeg, ← lift_umax.{w}, add_comm,
← (hs.sumElim_comp ht).lift_cardinalMk_eq_trdeg, mk_sum, lift_add, lift_lift, lift_lift]
@[stacks 030H] theorem trdeg_add_eq [Nontrivial R] {A : Type v} [CommRing A] [NoZeroDivisors A]
[Algebra R A] [Algebra S A] [FaithfulSMul R S] [FaithfulSMul S A] [IsScalarTower R S A] :
trdeg R S + trdeg S A = trdeg R A := by
rw [← (trdeg R S).lift_id, ← (trdeg S A).lift_id, ← (trdeg R A).lift_id]
exact lift_trdeg_add_eq R S A | theorem | RingTheory | [
"Mathlib.Combinatorics.Matroid.IndepAxioms",
"Mathlib.Combinatorics.Matroid.Rank.Cardinal",
"Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra",
"Mathlib.RingTheory.AlgebraicIndependent.Transcendental"
] | Mathlib/RingTheory/AlgebraicIndependent/TranscendenceBasis.lean | lift_trdeg_add_eq | null |
@[simp]
algebraicIndependent_unique_type_iff [Unique ι] :
AlgebraicIndependent R x ↔ Transcendental R (x default) := by
rw [transcendental_iff_injective, algebraicIndependent_iff_injective_aeval]
let i := (renameEquiv R (Equiv.equivPUnit.{_, 1} ι)).trans (pUnitAlgEquiv R)
have key : aeval (R := R) x = (Polynomial.aeval (R := R) (x default)).comp i := by
ext y
simp [i, Subsingleton.elim y default]
simp [key] | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | algebraicIndependent_unique_type_iff | A one-element family `x` is algebraically independent if and only if
its element is transcendental. |
algebraicIndependent_singleton_iff [Subsingleton ι] (i : ι) :
AlgebraicIndependent R x ↔ Transcendental R (x i) :=
letI := uniqueOfSubsingleton i
algebraicIndependent_unique_type_iff | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | algebraicIndependent_singleton_iff | null |
algebraicIndependent_iff_transcendental {x : A} :
AlgebraicIndependent R ![x] ↔ Transcendental R x := by
simp | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | algebraicIndependent_iff_transcendental | The one-element family `![x]` is algebraically independent if and only if
`x` is transcendental. |
transcendental (i : ι) : Transcendental R (x i) := by
have := hx.comp ![i] (Function.injective_of_subsingleton _)
have : AlgebraicIndependent R ![x i] := by rwa [← FinVec.map_eq] at this
rwa [← algebraicIndependent_iff_transcendental] | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | transcendental | If a family `x` is algebraically independent, then any of its element is transcendental. |
isEmpty_of_isAlgebraic [Algebra.IsAlgebraic R A] : IsEmpty ι := by
rcases isEmpty_or_nonempty ι with h | ⟨⟨i⟩⟩
· exact h
exact False.elim (hx.transcendental i (Algebra.IsAlgebraic.isAlgebraic _)) | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | isEmpty_of_isAlgebraic | If `A/R` is algebraic, then all algebraically independent families are empty. |
trdeg_eq_zero [Algebra.IsAlgebraic R A] : trdeg R A = 0 :=
bot_unique <| ciSup_le' fun s ↦ have := s.2.isEmpty_of_isAlgebraic; (Cardinal.mk_eq_zero _).le
variable (R A) in | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | trdeg_eq_zero | null |
trdeg_pos [Algebra.Transcendental R A] : 0 < trdeg R A :=
have ⟨x, hx⟩ := Algebra.Transcendental.transcendental (R := R) (A := A)
zero_lt_one.trans_le <| le_ciSup_of_le (Cardinal.bddAbove_range _)
⟨{x}, algebraicIndependent_unique_type_iff.mpr hx⟩ (by simp) | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | trdeg_pos | null |
trdeg_eq_zero_iff : trdeg R A = 0 ↔ Algebra.IsAlgebraic R A := by
by_cases h : Algebra.IsAlgebraic R A
· exact iff_of_true trdeg_eq_zero h
rw [← not_iff_not]
rw [← Algebra.transcendental_iff_not_isAlgebraic] at h ⊢
exact iff_of_true (trdeg_pos R A).ne' h | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | trdeg_eq_zero_iff | null |
trdeg_ne_zero_iff : trdeg R A ≠ 0 ↔ Algebra.Transcendental R A := by
rw [Algebra.transcendental_iff_not_isAlgebraic, Ne, trdeg_eq_zero_iff]
open AlgebraicIndependent | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | trdeg_ne_zero_iff | null |
AlgebraicIndependent.option_iff_transcendental (hx : AlgebraicIndependent R x) (a : A) :
AlgebraicIndependent R (fun o : Option ι ↦ o.elim a x) ↔
Transcendental (adjoin R (range x)) a := by
rw [algebraicIndependent_iff_injective_aeval, transcendental_iff_injective,
← AlgHom.coe_toRingHom, ← hx.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin,
RingHom.coe_comp]
exact Injective.of_comp_iff' (Polynomial.aeval a)
(mvPolynomialOptionEquivPolynomialAdjoin hx).bijective | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | AlgebraicIndependent.option_iff_transcendental | null |
AlgebraicIndependent.option_iff {a : A} :
AlgebraicIndependent R (fun o : Option ι ↦ o.elim a x) ↔
AlgebraicIndependent R x ∧ Transcendental (adjoin R (range x)) a :=
⟨fun h ↦ have := h.comp _ (Option.some_injective _); ⟨this,
(this.option_iff_transcendental _).mp h⟩, fun h ↦ (h.1.option_iff_transcendental _).mpr h.2⟩ | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | AlgebraicIndependent.option_iff | null |
AlgebraicIndepOn.insert_iff {s : Set ι} {i : ι} (h : i ∉ s) :
AlgebraicIndepOn R x (insert i s) ↔
AlgebraicIndepOn R x s ∧ Transcendental (adjoin R (x '' s)) (x i) := by
classical simp_rw [← algebraicIndependent_equiv (subtypeInsertEquivOption h).symm,
AlgebraicIndepOn]
convert option_iff (x := fun i : s ↦ x i) (a := x i) using 2
· ext (_ | _) <;> rfl
· rw [Set.image_eq_range] | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | AlgebraicIndepOn.insert_iff | null |
protected AlgebraicIndepOn.insert {s : Set ι} {i : ι} (hs : AlgebraicIndepOn R x s)
(hi : Transcendental (adjoin R (x '' s)) (x i)) : AlgebraicIndepOn R x (insert i s) := by
nontriviality R
have := hs.algebraMap_injective.nontrivial
exact (insert_iff fun h ↦ hi <| isAlgebraic_algebraMap
(⟨_, subset_adjoin ⟨i, h, rfl⟩⟩ : adjoin R (x '' s))).mpr ⟨hs, hi⟩ | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | AlgebraicIndepOn.insert | null |
algebraicIndependent_of_set_of_finite (s : Set ι)
(ind : AlgebraicIndependent R fun i : s ↦ x i)
(H : ∀ t : Set ι, t.Finite → AlgebraicIndependent R (fun i : t ↦ x i) →
∀ i ∉ s, i ∉ t → Transcendental (adjoin R (x '' t)) (x i)) :
AlgebraicIndependent R x := by
classical
refine algebraicIndependent_of_finite_type fun t hfin ↦ ?_
suffices AlgebraicIndependent R fun i : ↥(t ∩ s ∪ t \ s) ↦ x i from
this.comp (Equiv.setCongr (t.inter_union_diff s).symm) (Equiv.injective _)
refine hfin.diff.induction_on_subset _ (ind.comp (inclusion <| by simp) (inclusion_injective _))
fun {a u} ha hu ha' h ↦ ?_
have : a ∉ t ∩ s ∪ u := (·.elim (ha.2 ·.2) ha')
convert (((image_eq_range .. ▸ h.option_iff_transcendental <| x a).2 <| H _ (hfin.subset
(union_subset inter_subset_left <| hu.trans diff_subset)) h a ha.2 this).comp _
(subtypeInsertEquivOption this).injective).comp
(Equiv.setCongr union_insert) (Equiv.injective _) with x
by_cases h : ↑x = a <;> simp [h, Set.subtypeInsertEquivOption] | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | algebraicIndependent_of_set_of_finite | null |
algebraicIndependent_of_finite_type'
(hinj : Injective (algebraMap R A))
(H : ∀ t : Set ι, t.Finite → AlgebraicIndependent R (fun i : t ↦ x i) →
∀ i ∉ t, Transcendental (adjoin R (x '' t)) (x i)) :
AlgebraicIndependent R x :=
algebraicIndependent_of_set_of_finite ∅ (algebraicIndependent_empty_type_iff.mpr hinj)
fun t ht ind i _ ↦ H t ht ind i | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | algebraicIndependent_of_finite_type' | Variant of `algebraicIndependent_of_finite_type` using `Transcendental`. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.