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 ⌀ |
|---|---|---|---|---|---|---|
Algebra.IsIntegral.of_injective (f : A →ₐ[R] B) (hf : Function.Injective f)
[Algebra.IsIntegral R B] : Algebra.IsIntegral R A :=
⟨fun _ ↦ (isIntegral_algHom_iff f hf).mp (isIntegral _)⟩ | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | Algebra.IsIntegral.of_injective | null |
Algebra.IsIntegral.of_surjective [Algebra.IsIntegral R A]
(f : A →ₐ[R] B) (hf : Function.Surjective f) : Algebra.IsIntegral R B :=
isIntegral_def.mpr fun b ↦ let ⟨a, ha⟩ := hf b; ha ▸ (isIntegral_def.mp ‹_› a).map f | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | Algebra.IsIntegral.of_surjective | Homomorphic image of an integral algebra is an integral algebra. |
AlgEquiv.isIntegral_iff (e : A ≃ₐ[R] B) : Algebra.IsIntegral R A ↔ Algebra.IsIntegral R B :=
⟨fun h ↦ h.of_injective e.symm e.symm.injective, fun h ↦ h.of_injective e e.injective⟩ | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | AlgEquiv.isIntegral_iff | null |
Module.End.isIntegral {M : Type*} [AddCommGroup M] [Module R M] [Module.Finite R M] :
Algebra.IsIntegral R (Module.End R M) :=
⟨LinearMap.exists_monic_and_aeval_eq_zero R⟩
variable (R) in
@[nontriviality] | instance | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | Module.End.isIntegral | null |
IsIntegral.of_finite [Module.Finite R B] (x : B) : IsIntegral R x :=
(isIntegral_algHom_iff (Algebra.lmul R B) Algebra.lmul_injective).mp
(Algebra.IsIntegral.isIntegral _) | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | IsIntegral.of_finite | null |
isIntegral_of_noetherian (_ : IsNoetherian R B) (x : B) : IsIntegral R x :=
.of_finite R x
variable (R B) in | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | isIntegral_of_noetherian | null |
Algebra.IsIntegral.of_finite [Module.Finite R B] : Algebra.IsIntegral R B :=
⟨.of_finite R⟩ | instance | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | Algebra.IsIntegral.of_finite | null |
Algebra.isIntegral_of_surjective (H : Function.Surjective (algebraMap R B)) :
Algebra.IsIntegral R B :=
.of_surjective (Algebra.ofId R B) H | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | Algebra.isIntegral_of_surjective | null |
IsIntegral.of_mem_of_fg (S : Subalgebra R B)
(HS : S.toSubmodule.FG) (x : B) (hx : x ∈ S) : IsIntegral R x :=
have : Module.Finite R S := ⟨(fg_top _).mpr HS⟩
(isIntegral_algHom_iff S.val Subtype.val_injective).mpr (.of_finite R (⟨x, hx⟩ : S)) | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | IsIntegral.of_mem_of_fg | If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
then all elements of `S` are integral over `R`. |
isIntegral_of_submodule_noetherian (S : Subalgebra R B)
(H : IsNoetherian R (Subalgebra.toSubmodule S)) (x : B) (hx : x ∈ S) : IsIntegral R x :=
.of_mem_of_fg _ ((fg_top _).mp <| H.noetherian _) _ hx | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | isIntegral_of_submodule_noetherian | null |
isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R M] [Module A M]
[IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.FG)
(x : A) (hx : ∀ n ∈ N, x • n ∈ N) : IsIntegral R x := by
let A' : Subalgebra R A :=
{ carrier := { x | ∀ n ∈ N, x • n ∈ N }
mul_mem' := fun {a b} ha hb n hn => smul_smul a b n ▸ ha _ (hb _ hn)
one_mem' := fun n hn => (one_smul A n).symm ▸ hn
add_mem' := fun {a b} ha hb n hn => (add_smul a b n).symm ▸ N.add_mem (ha _ hn) (hb _ hn)
zero_mem' := fun n _hn => (zero_smul A n).symm ▸ N.zero_mem
algebraMap_mem' := fun r n hn => (algebraMap_smul A r n).symm ▸ N.smul_mem r hn }
let f : A' →ₐ[R] Module.End R N :=
AlgHom.ofLinearMap
{ toFun := fun x => (DistribMulAction.toLinearMap R M x).restrict x.prop
map_add' := by intro x y; ext; exact add_smul _ _ _
map_smul' := by intro r s; ext; apply smul_assoc }
(by ext; apply one_smul)
(by intro x y; ext; apply mul_smul)
obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by
by_contra! h'
apply hN
rwa [eq_bot_iff]
have : Function.Injective f := by
change Function.Injective f.toLinearMap
rw [← LinearMap.ker_eq_bot, eq_bot_iff]
intro s hs
have : s.1 • a = 0 := congr_arg Subtype.val (LinearMap.congr_fun hs ⟨a, ha₁⟩)
exact Subtype.ext ((eq_zero_or_eq_zero_of_smul_eq_zero this).resolve_right ha₂)
change IsIntegral R (A'.val ⟨x, hx⟩)
rw [isIntegral_algHom_iff A'.val Subtype.val_injective, ← isIntegral_algHom_iff f this]
haveI : Module.Finite R N := by rwa [Module.finite_def, Submodule.fg_top]
apply Algebra.IsIntegral.isIntegral
variable {f}
@[stacks 00GK] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | isIntegral_of_smul_mem_submodule | Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. |
RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
letI := f.toAlgebra
fun _ ↦ IsIntegral.of_mem_of_fg ⊤ h.1 _ trivial
alias RingHom.IsIntegral.of_finite := RingHom.Finite.to_isIntegral
variable (f) | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | RingHom.Finite.to_isIntegral | null |
RingHom.IsIntegralElem.of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
(hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z := by
letI : Algebra R S := f.toAlgebra
have := (IsIntegral.fg_adjoin_singleton hx).mul (IsIntegral.fg_adjoin_singleton hy)
rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
exact
IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z
(Algebra.mem_adjoin_iff.2 <| Subring.closure_mono Set.subset_union_right hz)
nonrec theorem IsIntegral.of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
(hz : z ∈ Subring.closure ({x, y} : Set A)) : IsIntegral R z :=
hx.of_mem_closure (algebraMap R A) hy hz
variable (f : R →+* B) | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | RingHom.IsIntegralElem.of_mem_closure | null |
RingHom.IsIntegralElem.add (f : R →+* S) {x y : S}
(hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
f.IsIntegralElem (x + y) :=
hx.of_mem_closure f hy <|
Subring.add_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl))
nonrec theorem IsIntegral.add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
IsIntegral R (x + y) :=
hx.add (algebraMap R A) hy
variable (f : R →+* S) | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | RingHom.IsIntegralElem.add | null |
RingHom.IsIntegralElem.neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
hx.of_mem_closure f hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl))) | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | RingHom.IsIntegralElem.neg | null |
RingHom.IsIntegralElem.of_neg {x : S} (h : f.IsIntegralElem (-x)) : f.IsIntegralElem x :=
neg_neg x ▸ h.neg
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | RingHom.IsIntegralElem.of_neg | null |
RingHom.IsIntegralElem.neg_iff {x : S} : f.IsIntegralElem (-x) ↔ f.IsIntegralElem x :=
⟨fun h => h.of_neg, fun h => h.neg⟩ | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | RingHom.IsIntegralElem.neg_iff | null |
IsIntegral.neg {x : B} (hx : IsIntegral R x) : IsIntegral R (-x) :=
.of_mem_of_fg _ hx.fg_adjoin_singleton _ (Subalgebra.neg_mem _ <| Algebra.subset_adjoin rfl) | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | IsIntegral.neg | null |
IsIntegral.of_neg {x : B} (hx : IsIntegral R (-x)) : IsIntegral R x :=
neg_neg x ▸ hx.neg
@[simp] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | IsIntegral.of_neg | null |
IsIntegral.neg_iff {x : B} : IsIntegral R (-x) ↔ IsIntegral R x :=
⟨IsIntegral.of_neg, IsIntegral.neg⟩ | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | IsIntegral.neg_iff | null |
RingHom.IsIntegralElem.sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
f.IsIntegralElem (x - y) := by
simpa only [sub_eq_add_neg] using hx.add f (hy.neg f)
nonrec theorem IsIntegral.sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
IsIntegral R (x - y) :=
hx.sub (algebraMap R A) hy | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | RingHom.IsIntegralElem.sub | null |
RingHom.IsIntegralElem.mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
f.IsIntegralElem (x * y) :=
hx.of_mem_closure f hy
(Subring.mul_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl)))
nonrec theorem IsIntegral.mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
IsIntegral R (x * y) :=
hx.mul (algebraMap R A) hy | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | RingHom.IsIntegralElem.mul | null |
IsIntegral.smul {R} [CommSemiring R] [Algebra R B] [Algebra S B] [Algebra R S]
[IsScalarTower R S B] {x : B} (r : R) (hx : IsIntegral S x) : IsIntegral S (r • x) :=
.of_mem_of_fg _ hx.fg_adjoin_singleton _ <| by
rw [← algebraMap_smul S]; apply Subalgebra.smul_mem; exact Algebra.subset_adjoin rfl
variable (R A) | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | IsIntegral.smul | null |
integralClosure : Subalgebra R A where
carrier := { r | IsIntegral R r }
zero_mem' := isIntegral_zero
one_mem' := isIntegral_one
add_mem' := IsIntegral.add
mul_mem' := IsIntegral.mul
algebraMap_mem' _ := isIntegral_algebraMap | def | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | integralClosure | The integral closure of `R` in an `R`-algebra `A`. |
mem_integralClosure_iff {a : A} : a ∈ integralClosure R A ↔ IsIntegral R a :=
Iff.rfl
variable {R} {A B : Type*} [Ring A] [Algebra R A] [Ring B] [Algebra R B] | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | mem_integralClosure_iff | null |
Algebra.IsIntegral.prod [Algebra.IsIntegral R A] [Algebra.IsIntegral R B] :
Algebra.IsIntegral R (A × B) :=
Algebra.isIntegral_def.mpr fun x ↦
(Algebra.isIntegral_def.mp ‹_› x.1).pair (Algebra.isIntegral_def.mp ‹_› x.2) | instance | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | Algebra.IsIntegral.prod | Product of two integral algebras is an integral algebra. |
IsIntegral.tmul [Ring B] [Algebra R A] [Algebra R B]
(x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x ⊗ₜ[R] y) := by
rw [← mul_one x, ← smul_eq_mul, ← smul_tmul']
exact smul _ (h.map_of_comp_eq (algebraMap R A)
(Algebra.TensorProduct.includeRight (R := R) (A := A) (B := B)).toRingHom
Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap)
variable (R A B) | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | IsIntegral.tmul | null |
Algebra.IsIntegral.tensorProduct [CommRing B]
[Algebra R A] [Algebra R B] [int : Algebra.IsIntegral R B] :
Algebra.IsIntegral A (A ⊗[R] B) where
isIntegral p := p.induction_on isIntegral_zero (fun _ s ↦ .tmul _ <| int.1 s) (fun _ _ ↦ .add) | instance | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | Algebra.IsIntegral.tensorProduct | null |
@[mk_iff] protected Algebra.IsIntegral : Prop where
isIntegral : ∀ x : A, IsIntegral R x
variable {R A} | class | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Defs.lean | Algebra.IsIntegral | An algebra is integral if every element of the extension is integral over the base ring. |
Algebra.isIntegral_def : Algebra.IsIntegral R A ↔ ∀ x : A, IsIntegral R x :=
⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩ | lemma | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Defs.lean | Algebra.isIntegral_def | null |
algebraMap_isIntegral_iff : (algebraMap R A).IsIntegral ↔ Algebra.IsIntegral R A :=
(Algebra.isIntegral_iff ..).symm | lemma | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Defs.lean | algebraMap_isIntegral_iff | null |
RingHom.isIntegralElem_map {x : R} : f.IsIntegralElem (f x) :=
⟨X - C x, monic_X_sub_C _, by simp⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | RingHom.isIntegralElem_map | null |
isIntegral_algebraMap {x : R} : IsIntegral R (algebraMap R A x) :=
(algebraMap R A).isIntegralElem_map | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | isIntegral_algebraMap | null |
IsIntegral.map {B C F : Type*} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
[IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B}
[FunLike F B C] [AlgHomClass F A B C] (f : F)
(hb : IsIntegral R b) : IsIntegral R (f b) := by
obtain ⟨P, hP⟩ := hb
refine ⟨P, hP.1, ?_⟩
rw [← aeval_def, ← aeval_map_algebraMap A,
aeval_algHom_apply, aeval_map_algebraMap, aeval_def, hP.2, map_zero] | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.map | null |
isIntegral_algHom_iff (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} :
IsIntegral R (f x) ↔ IsIntegral R x := by
refine ⟨fun ⟨p, hp, hx⟩ ↦ ⟨p, hp, ?_⟩, IsIntegral.map f⟩
rwa [← f.comp_algebraMap, ← AlgHom.coe_toRingHom, ← hom_eval₂, AlgHom.coe_toRingHom,
map_eq_zero_iff f hf] at hx | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | isIntegral_algHom_iff | null |
Submodule.span_range_natDegree_eq_adjoin {R A} [CommRing R] [Semiring A] [Algebra R A]
{x : A} {f : R[X]} (hf : f.Monic) (hfx : aeval x f = 0) :
span R (Finset.image (x ^ ·) (Finset.range (natDegree f))) =
Subalgebra.toSubmodule (Algebra.adjoin R {x}) := by
nontriviality A
have hf1 : f ≠ 1 := by rintro rfl; simp [one_ne_zero' A] at hfx
refine (span_le.mpr fun s hs ↦ ?_).antisymm fun r hr ↦ ?_
· rcases Finset.mem_image.1 hs with ⟨k, -, rfl⟩
exact (Algebra.adjoin R {x}).pow_mem (Algebra.subset_adjoin rfl) k
rw [Subalgebra.mem_toSubmodule, Algebra.adjoin_singleton_eq_range_aeval] at hr
rcases (aeval x).mem_range.mp hr with ⟨p, rfl⟩
rw [← modByMonic_add_div p hf, map_add, map_mul, hfx,
zero_mul, add_zero, ← sum_C_mul_X_pow_eq (p %ₘ f), aeval_def, eval₂_sum, sum_def]
refine sum_mem fun k hkq ↦ ?_
rw [C_mul_X_pow_eq_monomial, eval₂_monomial, ← Algebra.smul_def]
exact smul_mem _ _ (subset_span <| Finset.mem_image_of_mem _ <| Finset.mem_range.mpr <|
(le_natDegree_of_mem_supp _ hkq).trans_lt <| natDegree_modByMonic_lt p hf hf1) | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | Submodule.span_range_natDegree_eq_adjoin | null |
IsIntegral.fg_adjoin_singleton [Algebra R B] {x : B} (hx : IsIntegral R x) :
(Algebra.adjoin R {x}).toSubmodule.FG := by
classical
rcases hx with ⟨f, hfm, hfx⟩
use (Finset.range <| f.natDegree).image (x ^ ·)
exact span_range_natDegree_eq_adjoin hfm (by rwa [aeval_def])
variable (f : R →+* B) | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.fg_adjoin_singleton | null |
RingHom.isIntegralElem_zero : f.IsIntegralElem 0 :=
f.map_zero ▸ f.isIntegralElem_map | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | RingHom.isIntegralElem_zero | null |
isIntegral_zero [Algebra R B] : IsIntegral R (0 : B) :=
(algebraMap R B).isIntegralElem_zero | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | isIntegral_zero | null |
RingHom.isIntegralElem_one : f.IsIntegralElem 1 :=
f.map_one ▸ f.isIntegralElem_map | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | RingHom.isIntegralElem_one | null |
isIntegral_one [Algebra R B] : IsIntegral R (1 : B) :=
(algebraMap R B).isIntegralElem_one
variable (f : R →+* S) | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | isIntegral_one | null |
IsIntegral.of_pow [Algebra R B] {x : B} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
IsIntegral R x :=
have ⟨p, hmonic, heval⟩ := hx
⟨expand R n p, hmonic.expand hn, by rwa [← aeval_def, expand_aeval]⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.of_pow | null |
IsIntegral.of_aeval_monic {x : A} {p : R[X]} (monic : p.Monic)
(deg : p.natDegree ≠ 0) (hx : IsIntegral R (aeval x p)) : IsIntegral R x :=
have ⟨p, hmonic, heval⟩ := hx
⟨_, hmonic.comp monic deg, by rwa [eval₂_comp, ← aeval_def x]⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.of_aeval_monic | null |
IsIntegral.map_of_comp_eq {R S T U : Type*} [CommRing R] [Ring S]
[CommRing T] [Ring U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
(h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
IsIntegral T (ψ a) :=
let ⟨p, hp⟩ := ha
⟨p.map φ, hp.1.map _, by
rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.2, ψ.map_zero]⟩
@[simp] | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.map_of_comp_eq | null |
isIntegral_algEquiv {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
(f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
⟨fun h ↦ by simpa using h.map f.symm, IsIntegral.map f⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | isIntegral_algEquiv | null |
IsIntegral.tower_top [Algebra A B] [IsScalarTower R A B] {x : B}
(hx : IsIntegral R x) : IsIntegral A x :=
let ⟨p, hp, hpx⟩ := hx
⟨p.map <| algebraMap R A, hp.map _, by rw [← aeval_def, aeval_map_algebraMap, aeval_def, hpx]⟩
/- If `R` and `T` are isomorphic commutative rings and `S` is an `R`-algebra and a `T`-algebra in
a compatible way, then an element `a ∈ S` is integral over `R` if and only if it is integral
over `T`. -/ | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.tower_top | If `R → A → B` is an algebra tower,
then if the entire tower is an integral extension so is `A → B`. |
RingEquiv.isIntegral_iff {R S T : Type*} [CommRing R] [Ring S] [CommRing T]
[Algebra R S] [Algebra T S] (φ : R ≃+* T)
(h : (algebraMap T S).comp φ.toRingHom = algebraMap R S) (a : S) :
IsIntegral R a ↔ IsIntegral T a := by
constructor <;> intro ha
· letI : Algebra R T := φ.toRingHom.toAlgebra
letI : IsScalarTower R T S :=
⟨fun r t s ↦ by simp only [Algebra.smul_def, map_mul, ← h, mul_assoc]; rfl⟩
exact IsIntegral.tower_top ha
· have h' : (algebraMap T S) = (algebraMap R S).comp φ.symm.toRingHom := by
simp only [← h, RingHom.comp_assoc, RingEquiv.toRingHom_eq_coe, RingEquiv.comp_symm,
RingHomCompTriple.comp_eq]
letI : Algebra T R := φ.symm.toRingHom.toAlgebra
letI : IsScalarTower T R S :=
⟨fun r t s ↦ by simp only [Algebra.smul_def, map_mul, h', mul_assoc]; rfl⟩
exact IsIntegral.tower_top ha | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | RingEquiv.isIntegral_iff | null |
map_isIntegral_int {B C F : Type*} [Ring B] [Ring C] {b : B}
[FunLike F B C] [RingHomClass F B C] (f : F)
(hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
hb.map (f : B →+* C).toIntAlgHom | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | map_isIntegral_int | null |
IsIntegral.of_subring {x : B} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
hx.tower_top | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.of_subring | null |
protected IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A}
(h : IsIntegral R x) : IsIntegral R (algebraMap A B x) := by
rcases h with ⟨f, hf, hx⟩
use f, hf
rw [IsScalarTower.algebraMap_eq R A B, ← hom_eval₂, hx, RingHom.map_zero] | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.algebraMap | null |
isIntegral_algebraMap_iff [Algebra A B] [IsScalarTower R A B] {x : A}
(hAB : Function.Injective (algebraMap A B)) :
IsIntegral R (algebraMap A B x) ↔ IsIntegral R x :=
isIntegral_algHom_iff (IsScalarTower.toAlgHom R A B) hAB | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | isIntegral_algebraMap_iff | null |
isIntegral_iff_isIntegral_closure_finite {r : B} :
IsIntegral R r ↔ ∃ s : Set R, s.Finite ∧ IsIntegral (Subring.closure s) r := by
constructor <;> intro hr
· rcases hr with ⟨p, hmp, hpr⟩
refine ⟨_, Finset.finite_toSet _, p.restriction, monic_restriction.2 hmp, ?_⟩
rw [← aeval_def, ← aeval_map_algebraMap R r p.restriction, map_restriction, aeval_def, hpr]
rcases hr with ⟨s, _, hsr⟩
exact hsr.of_subring _
@[stacks 09GH] | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | isIntegral_iff_isIntegral_closure_finite | null |
fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
(Algebra.adjoin R s).toSubmodule.FG := by
induction s, hfs using Set.Finite.induction_on with
| empty =>
refine ⟨{1}, Submodule.ext fun x => ?_⟩
rw [Algebra.adjoin_empty, Finset.coe_singleton, ← one_eq_span, Algebra.toSubmodule_bot]
| @insert a s _ _ ih =>
rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule]
exact FG.mul
(ih fun i hi => his i <| Set.mem_insert_of_mem a hi)
(his a <| Set.mem_insert a s).fg_adjoin_singleton | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | fg_adjoin_of_finite | null |
Algebra.finite_adjoin_of_finite_of_isIntegral {s : Set A} (hf : s.Finite)
(hi : ∀ x ∈ s, IsIntegral R x) : Module.Finite R (adjoin R s) :=
Module.Finite.iff_fg.mpr <| fg_adjoin_of_finite hf hi | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | Algebra.finite_adjoin_of_finite_of_isIntegral | null |
Algebra.finite_adjoin_simple_of_isIntegral {x : B} (hi : IsIntegral R x) :
Module.Finite R (adjoin R {x}) :=
Module.Finite.iff_fg.mpr hi.fg_adjoin_singleton | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | Algebra.finite_adjoin_simple_of_isIntegral | null |
isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
(hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (s : Set A)) :=
isNoetherian_of_fg_of_noetherian _ (fg_adjoin_of_finite s.finite_toSet hs) | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | isNoetherian_adjoin_finset | null |
IsIntegral.pair {x : A × B} (hx₁ : IsIntegral R x.1) (hx₂ : IsIntegral R x.2) :
IsIntegral R x := by
obtain ⟨p₁, ⟨hp₁Monic, hp₁Eval⟩⟩ := hx₁
obtain ⟨p₂, ⟨hp₂Monic, hp₂Eval⟩⟩ := hx₂
refine ⟨p₁ * p₂, ⟨hp₁Monic.mul hp₂Monic, ?_⟩⟩
rw [← aeval_def] at *
rw [aeval_prod_apply, aeval_mul, hp₁Eval, zero_mul, aeval_mul, hp₂Eval, mul_zero,
Prod.zero_eq_mk] | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.pair | An element of a product algebra is integral if each component is integral. |
IsIntegral.pair_iff {x : A × B} : IsIntegral R x ↔ IsIntegral R x.1 ∧ IsIntegral R x.2 :=
⟨fun h ↦ ⟨h.map (AlgHom.fst R A B), h.map (AlgHom.snd R A B)⟩, fun h ↦ h.1.pair h.2⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs",
"Mathlib.Algebra.Polynomial.Expand",
"Mathlib.RingTheory.Adjoin.Polynomial",
"Mathlib.RingTheory.Finiteness.Subalgebra",
"Mathlib.RingTheory.Polynomial.Tower"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean | IsIntegral.pair_iff | An element of a product algebra is integral iff each component is integral. |
RingHom.IsIntegralElem (f : R →+* A) (x : A) :=
∃ p : R[X], Monic p ∧ eval₂ f x p = 0 | def | RingTheory | [
"Mathlib.Algebra.Polynomial.Degree.Definitions",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.lean | RingHom.IsIntegralElem | An element `x` of `A` is said to be integral over `R` with respect to `f`
if it is a root of a monic polynomial `p : R[X]` evaluated under `f` |
@[algebraize Algebra.IsIntegral.mk, stacks 00GI "(2)"]
RingHom.IsIntegral (f : R →+* A) :=
∀ x : A, f.IsIntegralElem x
variable [Algebra R A] (R) | def | RingTheory | [
"Mathlib.Algebra.Polynomial.Degree.Definitions",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.lean | RingHom.IsIntegral | A ring homomorphism `f : R →+* A` is said to be integral
if every element `A` is integral with respect to the map `f` |
IsIntegral (x : A) : Prop :=
(algebraMap R A).IsIntegralElem x | def | RingTheory | [
"Mathlib.Algebra.Polynomial.Degree.Definitions",
"Mathlib.Algebra.Polynomial.Eval.Defs",
"Mathlib.Tactic.Algebraize"
] | Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.lean | IsIntegral | An element `x` of an algebra `A` over a commutative ring `R` is said to be *integral*,
if it is a root of some monic polynomial `p : R[X]`.
Equivalently, the element is integral over `R` with respect to the induced `algebraMap` |
IsIntegral.isUnit [Field R] [Ring S] [IsDomain S] [Algebra R S] {x : S}
(int : IsIntegral R x) (h0 : x ≠ 0) : IsUnit x :=
have : FiniteDimensional R (adjoin R {x}) := ⟨(Submodule.fg_top _).mpr int.fg_adjoin_singleton⟩
(FiniteDimensional.isUnit R (K := adjoin R {x})
(x := ⟨x, subset_adjoin rfl⟩) <| mt Subtype.ext_iff.mp h0).map (adjoin R {x}).val | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.isUnit | A nonzero element in a domain integral over a field is a unit. |
isField_of_isIntegral_of_isField' [CommRing R] [CommRing S] [IsDomain S]
[Algebra R S] [Algebra.IsIntegral R S] (hR : IsField R) : IsField S where
exists_pair_ne := ⟨0, 1, zero_ne_one⟩
mul_comm := mul_comm
mul_inv_cancel {x} hx := by
letI := hR.toField
obtain ⟨y, rfl⟩ := (Algebra.IsIntegral.isIntegral (R := R) x).isUnit hx
exact ⟨y.inv, y.val_inv⟩
variable [Field R] [DivisionRing S] [Algebra R S] {x : S} {A : Subalgebra R S} | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | isField_of_isIntegral_of_isField' | A commutative domain that is an integral algebra over a field is a field. |
IsIntegral.inv_mem_adjoin (int : IsIntegral R x) : x⁻¹ ∈ adjoin R {x} := by
obtain rfl | h0 := eq_or_ne x 0
· rw [inv_zero]; exact Subalgebra.zero_mem _
have : FiniteDimensional R (adjoin R {x}) := ⟨(Submodule.fg_top _).mpr int.fg_adjoin_singleton⟩
obtain ⟨⟨y, hy⟩, h1⟩ := FiniteDimensional.exists_mul_eq_one R
(K := adjoin R {x}) (x := ⟨x, subset_adjoin rfl⟩) (mt Subtype.ext_iff.mp h0)
rwa [← mul_left_cancel₀ h0 ((Subtype.ext_iff.mp h1).trans (mul_inv_cancel₀ h0).symm)] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.inv_mem_adjoin | null |
IsIntegral.inv_mem (int : IsIntegral R x) (hx : x ∈ A) : x⁻¹ ∈ A :=
adjoin_le (Set.singleton_subset_iff.mpr hx) int.inv_mem_adjoin | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.inv_mem | The inverse of an integral element in a subalgebra of a division ring over a field
also lies in that subalgebra. |
Algebra.IsIntegral.inv_mem [Algebra.IsIntegral R A] (hx : x ∈ A) : x⁻¹ ∈ A :=
((isIntegral_algHom_iff A.val Subtype.val_injective).mpr <|
Algebra.IsIntegral.isIntegral (⟨x, hx⟩ : A)).inv_mem hx | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | Algebra.IsIntegral.inv_mem | An integral subalgebra of a division ring over a field is closed under inverses. |
IsIntegral.inv (int : IsIntegral R x) : IsIntegral R x⁻¹ :=
.of_mem_of_fg _ int.fg_adjoin_singleton _ int.inv_mem_adjoin | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.inv | The inverse of an integral element in a division ring over a field is also integral. |
IsIntegral.mem_of_inv_mem (int : IsIntegral R x) (inv_mem : x⁻¹ ∈ A) : x ∈ A := by
rw [← inv_inv x]; exact int.inv.inv_mem inv_mem | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.mem_of_inv_mem | null |
Algebra.IsIntegral.finite [Algebra.IsIntegral R A] [h' : Algebra.FiniteType R A] :
Module.Finite R A :=
have ⟨s, hs⟩ := h'
⟨by apply hs ▸ fg_adjoin_of_finite s.finite_toSet fun x _ ↦ Algebra.IsIntegral.isIntegral x⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | Algebra.IsIntegral.finite | The [Kurosh problem](https://en.wikipedia.org/wiki/Kurosh_problem) asks to show that
this is still true when `A` is not necessarily commutative and `R` is a field, but it has
been solved in the negative. See https://arxiv.org/pdf/1706.02383.pdf for criteria for a
finitely generated algebraic (= integral) algebra over a field to be finite dimensional.
This could be an `instance`, but we tend to go from `Module.Finite` to `IsIntegral`/`IsAlgebraic`,
and making it an instance will cause the search to be complicated a lot. |
Algebra.finite_iff_isIntegral_and_finiteType :
Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A :=
⟨fun _ ↦ ⟨⟨.of_finite R⟩, inferInstance⟩, fun ⟨h, _⟩ ↦ h.finite⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | Algebra.finite_iff_isIntegral_and_finiteType | finite = integral + finite type |
RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite :=
let _ := f.toAlgebra
let _ : Algebra.IsIntegral R S := ⟨h⟩
Algebra.IsIntegral.finite (h' := h')
alias RingHom.Finite.of_isIntegral_of_finiteType := RingHom.IsIntegral.to_finite | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | RingHom.IsIntegral.to_finite | null |
RingHom.finite_iff_isIntegral_and_finiteType : f.Finite ↔ f.IsIntegral ∧ f.FiniteType :=
⟨fun h ↦ ⟨h.to_isIntegral, h.to_finiteType⟩, fun ⟨h, h'⟩ ↦ h.to_finite h'⟩
variable (f : R →+* S) (R A) | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | RingHom.finite_iff_isIntegral_and_finiteType | finite = integral + finite type |
mem_integralClosure_iff_mem_fg {r : A} :
r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
⟨fun hr =>
⟨Algebra.adjoin R {r}, hr.fg_adjoin_singleton, Algebra.subset_adjoin rfl⟩,
fun ⟨M, Hf, hrM⟩ => .of_mem_of_fg M Hf _ hrM⟩
variable {R A} | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | mem_integralClosure_iff_mem_fg | null |
adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
Algebra.adjoin R {x} ≤ integralClosure R A := by
rw [Algebra.adjoin_le_iff]
simp only [SetLike.mem_coe, Set.singleton_subset_iff]
exact hx | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | adjoin_le_integralClosure | null |
le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
S ≤ integralClosure R A ↔ Algebra.IsIntegral R S :=
SetLike.forall.symm.trans <|
(forall_congr' fun x =>
show IsIntegral R (algebraMap S A x) ↔ IsIntegral R x from
isIntegral_algebraMap_iff Subtype.coe_injective).trans
Algebra.isIntegral_def.symm | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | le_integralClosure_iff_isIntegral | null |
Algebra.IsIntegral.adjoin {S : Set A} (hS : ∀ x ∈ S, IsIntegral R x) :
Algebra.IsIntegral R (adjoin R S) :=
le_integralClosure_iff_isIntegral.mp <| adjoin_le hS | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | Algebra.IsIntegral.adjoin | null |
integralClosure_eq_top_iff : integralClosure R A = ⊤ ↔ Algebra.IsIntegral R A := by
rw [← top_le_iff, le_integralClosure_iff_isIntegral,
(Subalgebra.topEquiv (R := R) (A := A)).isIntegral_iff] -- explicit arguments for speedup | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | integralClosure_eq_top_iff | null |
Algebra.isIntegral_sup {S T : Subalgebra R A} :
Algebra.IsIntegral R (S ⊔ T : Subalgebra R A) ↔
Algebra.IsIntegral R S ∧ Algebra.IsIntegral R T := by
simp_rw [← le_integralClosure_iff_isIntegral, sup_le_iff] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | Algebra.isIntegral_sup | null |
Algebra.isIntegral_iSup {ι} (S : ι → Subalgebra R A) :
Algebra.IsIntegral R ↑(iSup S) ↔ ∀ i, Algebra.IsIntegral R (S i) := by
simp_rw [← le_integralClosure_iff_isIntegral, iSup_le_iff] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | Algebra.isIntegral_iSup | null |
integralClosure_map_algEquiv [Algebra R S] (f : A ≃ₐ[R] S) :
(integralClosure R A).map (f : A →ₐ[R] S) = integralClosure R S := by
ext y
rw [Subalgebra.mem_map]
constructor
· rintro ⟨x, hx, rfl⟩
exact hx.map f
· intro hy
use f.symm y, hy.map (f.symm : S →ₐ[R] A)
simp | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | integralClosure_map_algEquiv | Mapping an integral closure along an `AlgEquiv` gives the integral closure. |
AlgHom.mapIntegralClosure [Algebra R S] (f : A →ₐ[R] S) :
integralClosure R A →ₐ[R] integralClosure R S :=
(f.restrictDomain (integralClosure R A)).codRestrict (integralClosure R S) (fun ⟨_, h⟩ => h.map f)
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | AlgHom.mapIntegralClosure | An `AlgHom` between two rings restrict to an `AlgHom` between the integral closures inside
them. |
AlgHom.coe_mapIntegralClosure [Algebra R S] (f : A →ₐ[R] S)
(x : integralClosure R A) : (f.mapIntegralClosure x : S) = f (x : A) := rfl | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | AlgHom.coe_mapIntegralClosure | null |
AlgEquiv.mapIntegralClosure [Algebra R S] (f : A ≃ₐ[R] S) :
integralClosure R A ≃ₐ[R] integralClosure R S :=
AlgEquiv.ofAlgHom (f : A →ₐ[R] S).mapIntegralClosure (f.symm : S →ₐ[R] A).mapIntegralClosure
(AlgHom.ext fun _ ↦ Subtype.ext (f.right_inv _))
(AlgHom.ext fun _ ↦ Subtype.ext (f.left_inv _))
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | AlgEquiv.mapIntegralClosure | An `AlgEquiv` between two rings restrict to an `AlgEquiv` between the integral closures inside
them. |
AlgEquiv.coe_mapIntegralClosure [Algebra R S] (f : A ≃ₐ[R] S)
(x : integralClosure R A) : (f.mapIntegralClosure x : S) = f (x : A) := rfl | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | AlgEquiv.coe_mapIntegralClosure | null |
integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
let ⟨p, hpm, hpx⟩ := x.2
⟨p, hpm,
Subtype.eq <| by
rwa [← aeval_def, ← Subalgebra.val_apply, aeval_algHom_apply] at hpx⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | integralClosure.isIntegral | null |
integralClosure.AlgebraIsIntegral : Algebra.IsIntegral R (integralClosure R A) :=
⟨integralClosure.isIntegral⟩ | instance | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | integralClosure.AlgebraIsIntegral | null |
IsIntegral.of_mul_unit {x y : B} {r : R} (hr : algebraMap R B r * y = 1)
(hx : IsIntegral R (x * y)) : IsIntegral R x := by
obtain ⟨p, p_monic, hp⟩ := hx
refine ⟨scaleRoots p r, (monic_scaleRoots_iff r).2 p_monic, ?_⟩
convert scaleRoots_aeval_eq_zero hp
rw [Algebra.commutes] at hr ⊢
rw [mul_assoc, hr, mul_one]; rfl | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.of_mul_unit | null |
RingHom.IsIntegralElem.of_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
(hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x :=
letI : Algebra R S := f.toAlgebra
IsIntegral.of_mul_unit hr hx | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | RingHom.IsIntegralElem.of_mul_unit | null |
IsIntegral.of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
∀ x ∈ Subring.closure G, IsIntegral R x := fun _ hx ↦
Subring.closure_induction hG isIntegral_zero isIntegral_one (fun _ _ _ _ ↦ IsIntegral.add)
(fun _ _ ↦ IsIntegral.neg) (fun _ _ _ _ ↦ IsIntegral.mul) hx | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.of_mem_closure' | Generalization of `IsIntegral.of_mem_closure` bootstrapped up from that lemma |
IsIntegral.of_mem_closure'' {S : Type*} [CommRing S] {f : R →+* S} (G : Set S)
(hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x := fun x hx =>
@IsIntegral.of_mem_closure' R S _ _ f.toAlgebra G hG x hx | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.of_mem_closure'' | null |
IsIntegral.pow {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n) :=
.of_mem_of_fg _ h.fg_adjoin_singleton _ <|
Subalgebra.pow_mem _ (by exact Algebra.subset_adjoin rfl) _ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.pow | null |
IsIntegral.nsmul {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (n • x) :=
h.smul n | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.nsmul | null |
IsIntegral.zsmul {x : B} (h : IsIntegral R x) (n : ℤ) : IsIntegral R (n • x) :=
h.smul n | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.zsmul | null |
IsIntegral.multiset_prod {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) :
IsIntegral R s.prod :=
(integralClosure R A).multiset_prod_mem h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.multiset_prod | null |
IsIntegral.multiset_sum {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) :
IsIntegral R s.sum :=
(integralClosure R A).multiset_sum_mem h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.multiset_sum | null |
IsIntegral.prod {α : Type*} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
IsIntegral R (∏ x ∈ s, f x) :=
(integralClosure R A).prod_mem h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.prod | null |
IsIntegral.sum {α : Type*} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
IsIntegral R (∑ x ∈ s, f x) :=
(integralClosure R A).sum_mem h | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.sum | null |
IsIntegral.det {n : Type*} [Fintype n] [DecidableEq n] {M : Matrix n n A}
(h : ∀ i j, IsIntegral R (M i j)) : IsIntegral R M.det := by
rw [Matrix.det_apply]
exact IsIntegral.sum _ fun σ _hσ ↦ (IsIntegral.prod _ fun i _hi => h _ _).zsmul _
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.det | null |
IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n) ↔ IsIntegral R x :=
⟨IsIntegral.of_pow hn, fun hx ↦ hx.pow n⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | IsIntegral.pow_iff | null |
Algebra.IsPushout.isIntegral' [IsPushout R A S SA] : Algebra.IsIntegral A SA :=
(equiv R A S SA).isIntegral_iff.mp inferInstance | theorem | RingTheory | [
"Mathlib.Algebra.Polynomial.Roots",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.IntegralClosure.Algebra.Basic",
"Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.Ten... | Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean | Algebra.IsPushout.isIntegral' | null |
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