Integral closure of a subring.

If A is an R-algebra then a : A is integral over R if it is a root of a monic polynomial with coefficients in R. Enough theory is developed to prove that integral elements form a sub-R-algebra of A.

Main definitions

Let R be a CommRing and let A be an R-algebra.

• RingHom.IsIntegralElem (f : R →+* A) (x : A) : x is integral with respect to the map f,

• IsIntegral (x : A) : x is integral over R, i.e., is a root of a monic polynomial with coefficients in R.

• integralClosure R A : the integral closure of R in A, regarded as a sub-R-algebra of A.

def RingHom.IsIntegralElem {R : Type u_1} {A : Type u_3} [CommRing R] [Ring A] (f : R →+* A) (x : A) : Prop

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

Equations

• f.IsIntegralElem x = ∃ (p : Polynomial R), p.Monic ∧ Polynomial.eval₂ f x p = 0

def RingHom.IsIntegral {R : Type u_1} {A : Type u_3} [CommRing R] [Ring A] (f : R →+* A) : Prop

A ring homomorphism f : R →+* A is said to be integral if every element A is integral with respect to the map f

Equations

• f.IsIntegral = ∀ (x : A), f.IsIntegralElem x

def IsIntegral (R : Type u_1) {A : Type u_3} [CommRing R] [Ring A] [Algebra R A] (x : A) : Prop

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

Equations

• IsIntegral R x = (algebraMap R A).IsIntegralElem x

def Algebra.IsIntegral (R : Type u_1) (A : Type u_3) [CommRing R] [Ring A] [Algebra R A] : Prop

An algebra is integral if every element of the extension is integral over the base ring

Equations

• Algebra.IsIntegral R A = ∀ (x : A), IsIntegral R x

theorem RingHom.isIntegralElem_map {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] (f : R →+* S) {x : R} : f.IsIntegralElem (f x)

theorem isIntegral_algebraMap {R : Type u_1} {A : Type u_3} [CommRing R] [Ring A] [Algebra R A] {x : R} : IsIntegral R ((algebraMap R A) x)

theorem IsIntegral.map {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {B : Type u_5} {C : Type u_6} {F : Type u_7} [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)

theorem IsIntegral.map_of_comp_eq {R : Type u_5} {S : Type u_6} {T : Type u_7} {U : Type u_8} [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)

theorem isIntegral_algHom_iff {R : Type u_1} [CommRing R] {A : Type u_5} {B : Type u_6} [Ring A] [Ring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x

theorem Algebra.IsIntegral.of_injective {R : Type u_1} [CommRing R] {A : Type u_5} {B : Type u_6} [Ring A] [Ring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) (hf : Function.Injective ⇑f) (h : Algebra.IsIntegral R B) : Algebra.IsIntegral R A

@[simp]

theorem isIntegral_algEquiv {R : Type u_1} [CommRing R] {A : Type u_5} {B : Type u_6} [Ring A] [Ring B] [Algebra R A] [Algebra R B] (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x

theorem IsIntegral.tower_top {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] {x : B} (hx : IsIntegral R x) : IsIntegral A x

If R → A → B is an algebra tower, then if the entire tower is an integral extension so is A → B.

theorem map_isIntegral_int {B : Type u_5} {C : Type u_6} {F : Type u_7} [Ring B] [Ring C] {b : B} [FunLike F B C] [RingHomClass F B C] (f : F) (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b)

theorem IsIntegral.of_subring {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (T : Subring R) (hx : IsIntegral (↥T) x) : IsIntegral R x

theorem IsIntegral.algebraMap {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] {x : A} (h : IsIntegral R x) : IsIntegral R ((algebraMap A B) x)

theorem isIntegral_algebraMap_iff {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] {x : A} (hAB : Function.Injective ⇑(algebraMap A B)) : IsIntegral R ((algebraMap A B) x) ↔ IsIntegral R x

theorem isIntegral_iff_isIntegral_closure_finite {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {r : B} : IsIntegral R r ↔ ∃ (s : Set R), s.Finite ∧ IsIntegral (↥(Subring.closure s)) r

theorem Submodule.span_range_natDegree_eq_adjoin {R : Type u_5} {A : Type u_6} [CommRing R] [Semiring A] [Algebra R A] {x : A} {f : Polynomial R} (hf : f.Monic) (hfx : (Polynomial.aeval x) f = 0) : Submodule.span R ↑(Finset.image (fun (x_1 : ℕ) => x ^ x_1) (Finset.range f.natDegree)) = Subalgebra.toSubmodule (Algebra.adjoin R {x})

theorem IsIntegral.fg_adjoin_singleton {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (hx : IsIntegral R x) : (Subalgebra.toSubmodule (Algebra.adjoin R {x})).FG

theorem fg_adjoin_of_finite {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) : (Subalgebra.toSubmodule (Algebra.adjoin R s)).FG

theorem isNoetherian_adjoin_finset {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] [IsNoetherianRing R] (s : Finset A) (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R ↥(Algebra.adjoin R ↑s)

theorem Module.End.isIntegral {R : Type u_1} [CommRing R] {M : Type u_5} [AddCommGroup M] [Module R M] [Module.Finite R M] : Algebra.IsIntegral R (Module.End R M)

theorem IsIntegral.of_finite (R : Type u_1) {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] [Module.Finite R B] (x : B) : IsIntegral R x

theorem Algebra.IsIntegral.of_finite (R : Type u_1) (B : Type u_3) [CommRing R] [Ring B] [Algebra R B] [Module.Finite R B] : Algebra.IsIntegral R B

theorem IsIntegral.of_mem_of_fg {R : Type u_1} [CommRing R] {A : Type u_5} [Ring A] [Algebra R A] (S : Subalgebra R A) (HS : (Subalgebra.toSubmodule S).FG) (x : A) (hx : x ∈ S) : IsIntegral R x

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.

theorem isIntegral_of_noetherian {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] : IsNoetherian R B → ∀ (x : B), IsIntegral R x

theorem isIntegral_of_submodule_noetherian {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] (S : Subalgebra R B) (H : IsNoetherian R ↥(Subalgebra.toSubmodule S)) (x : B) (hx : x ∈ S) : IsIntegral R x

theorem isIntegral_of_smul_mem_submodule {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {M : Type u_5} [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

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.

theorem RingHom.Finite.to_isIntegral {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} (h : f.Finite) : f.IsIntegral

theorem RingHom.IsIntegral.of_finite {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} (h : f.Finite) : f.IsIntegral

Alias of RingHom.Finite.to_isIntegral.

theorem Algebra.IsIntegral.finite {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (h : Algebra.IsIntegral R A) [h' : Algebra.FiniteType R A] : Module.Finite R A

The 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.

theorem Algebra.finite_iff_isIntegral_and_finiteType {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] : Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A

finite = integral + finite type

theorem RingHom.IsIntegral.to_finite {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite

theorem RingHom.Finite.of_isIntegral_of_finiteType {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite

Alias of RingHom.IsIntegral.to_finite.

theorem RingHom.finite_iff_isIntegral_and_finiteType {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] {f : R →+* S} : f.Finite ↔ f.IsIntegral ∧ f.FiniteType

finite = integral + finite type

theorem RingHom.IsIntegralElem.of_mem_closure {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {x : S} {y : S} {z : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure {x, y}) : f.IsIntegralElem z

theorem IsIntegral.of_mem_closure {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {x : A} {y : A} {z : A} (hx : IsIntegral R x) (hy : IsIntegral R y) (hz : z ∈ Subring.closure {x, y}) : IsIntegral R z

theorem RingHom.isIntegralElem_zero {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] (f : R →+* B) : f.IsIntegralElem 0

theorem isIntegral_zero {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] : IsIntegral R 0

theorem RingHom.isIntegralElem_one {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] (f : R →+* B) : f.IsIntegralElem 1

theorem isIntegral_one {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] : IsIntegral R 1

theorem RingHom.IsIntegralElem.add {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {x : S} {y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) : f.IsIntegralElem (x + y)

theorem IsIntegral.add {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {x : A} {y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) : IsIntegral R (x + y)

theorem RingHom.IsIntegralElem.neg {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x)

theorem IsIntegral.neg {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (hx : IsIntegral R x) : IsIntegral R (-x)

theorem RingHom.IsIntegralElem.sub {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {x : S} {y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) : f.IsIntegralElem (x - y)

theorem IsIntegral.sub {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {x : A} {y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) : IsIntegral R (x - y)

theorem RingHom.IsIntegralElem.mul {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {x : S} {y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) : f.IsIntegralElem (x * y)

theorem IsIntegral.mul {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {x : A} {y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) : IsIntegral R (x * y)

theorem IsIntegral.smul {B : Type u_3} {S : Type u_4} [Ring B] {R : Type u_5} [CommSemiring R] [CommRing S] [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)

theorem IsIntegral.of_pow {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R (x ^ n)) : IsIntegral R x

def integralClosure (R : Type u_1) (A : Type u_2) [CommRing R] [CommRing A] [Algebra R A] : Subalgebra R A

The integral closure of R in an R-algebra A.

Equations

• integralClosure R A = { carrier := {r : A | IsIntegral R r}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, algebraMap_mem' := ⋯ }

theorem mem_integralClosure_iff_mem_fg (R : Type u_1) (A : Type u_2) [CommRing R] [CommRing A] [Algebra R A] {r : A} : r ∈ integralClosure R A ↔ ∃ (M : Subalgebra R A), (Subalgebra.toSubmodule M).FG ∧ r ∈ M

theorem adjoin_le_integralClosure {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {x : A} (hx : IsIntegral R x) : Algebra.adjoin R {x} ≤ integralClosure R A

theorem le_integralClosure_iff_isIntegral {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {S : Subalgebra R A} : S ≤ integralClosure R A ↔ Algebra.IsIntegral R ↥S

theorem Algebra.isIntegral_sup {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {S : Subalgebra R A} {T : Subalgebra R A} : Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R ↥S ∧ Algebra.IsIntegral R ↥T

theorem integralClosure_map_algEquiv {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A ≃ₐ[R] S) : Subalgebra.map (↑f) (integralClosure R A) = integralClosure R S

Mapping an integral closure along an AlgEquiv gives the integral closure.

def AlgHom.mapIntegralClosure {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A →ₐ[R] S) : ↥(integralClosure R A) →ₐ[R] ↥(integralClosure R S)

An AlgHom between two rings restrict to an AlgHom between the integral closures inside them.

Equations

• f.mapIntegralClosure = (AlgHom.restrictDomain (↥(integralClosure R A)) f).codRestrict (integralClosure R S) ⋯

@[simp]

theorem AlgHom.coe_mapIntegralClosure {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A →ₐ[R] S) (x : ↥(integralClosure R A)) : ↑(f.mapIntegralClosure x) = f ↑x

def AlgEquiv.mapIntegralClosure {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A ≃ₐ[R] S) : ↥(integralClosure R A) ≃ₐ[R] ↥(integralClosure R S)

An AlgEquiv between two rings restrict to an AlgEquiv between the integral closures inside them.

Equations

• f.mapIntegralClosure = AlgEquiv.ofAlgHom (↑f).mapIntegralClosure (↑f.symm).mapIntegralClosure ⋯ ⋯

@[simp]

theorem AlgEquiv.coe_mapIntegralClosure {R : Type u_1} {A : Type u_2} {S : Type u_4} [CommRing R] [CommRing A] [CommRing S] [Algebra R A] [Algebra R S] (f : A ≃ₐ[R] S) (x : ↥(integralClosure R A)) : ↑(f.mapIntegralClosure x) = f ↑x

theorem integralClosure.isIntegral {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (x : ↥(integralClosure R A)) : IsIntegral R x

theorem IsIntegral.of_mul_unit {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} {y : B} {r : R} (hr : (algebraMap R B) r * y = 1) (hx : IsIntegral R (x * y)) : IsIntegral R x

theorem RingHom.IsIntegralElem.of_mul_unit {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (x : S) (y : S) (r : R) (hr : f r * y = 1) (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x

theorem IsIntegral.of_mem_closure' {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) (x : A) : x ∈ Subring.closure G → IsIntegral R x

Generalization of IsIntegral.of_mem_closure bootstrapped up from that lemma

theorem IsIntegral.of_mem_closure'' {R : Type u_1} [CommRing R] {S : Type u_5} [CommRing S] {f : R →+* S} (G : Set S) (hG : ∀ x ∈ G, f.IsIntegralElem x) (x : S) : x ∈ Subring.closure G → f.IsIntegralElem x

theorem IsIntegral.pow {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n)

theorem IsIntegral.nsmul {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (n • x)

theorem IsIntegral.zsmul {R : Type u_1} {B : Type u_3} [CommRing R] [Ring B] [Algebra R B] {x : B} (h : IsIntegral R x) (n : ℤ) : IsIntegral R (n • x)

theorem IsIntegral.multiset_prod {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) : IsIntegral R s.prod

theorem IsIntegral.multiset_sum {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) : IsIntegral R s.sum

theorem IsIntegral.prod {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {α : Type u_5} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) : IsIntegral R (s.prod fun (x : α) => f x)

theorem IsIntegral.sum {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {α : Type u_5} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) : IsIntegral R (s.sum fun (x : α) => f x)

theorem IsIntegral.det {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {n : Type u_5} [Fintype n] [DecidableEq n] {M : Matrix n n A} (h : ∀ (i j : n), IsIntegral R (M i j)) : IsIntegral R M.det

@[simp]

theorem IsIntegral.pow_iff {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n) ↔ IsIntegral R x

theorem IsIntegral.tmul {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra R A] [Algebra R B] (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x ⊗ₜ[R] y)

noncomputable def normalizeScaleRoots {R : Type u_1} [CommRing R] (p : Polynomial R) : Polynomial R

The monic polynomial whose roots are p.leadingCoeff * x for roots x of p.

Equations

• normalizeScaleRoots p = p.support.sum fun (i : ℕ) => (Polynomial.monomial i) (if i = p.natDegree then 1 else p.coeff i * p.leadingCoeff ^ (p.natDegree - 1 - i))

theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow {R : Type u_1} [CommRing R] (p : Polynomial R) (i : ℕ) (hp : 1 ≤ p.natDegree) : (normalizeScaleRoots p).coeff i * p.leadingCoeff ^ i = p.coeff i * p.leadingCoeff ^ (p.natDegree - 1)

theorem leadingCoeff_smul_normalizeScaleRoots {R : Type u_1} [CommRing R] (p : Polynomial R) : p.leadingCoeff • normalizeScaleRoots p = p.scaleRoots p.leadingCoeff

theorem normalizeScaleRoots_support {R : Type u_1} [CommRing R] (p : Polynomial R) : (normalizeScaleRoots p).support ≤ p.support

theorem normalizeScaleRoots_degree {R : Type u_1} [CommRing R] (p : Polynomial R) : (normalizeScaleRoots p).degree = p.degree

theorem normalizeScaleRoots_eval₂_leadingCoeff_mul {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (p : Polynomial R) (h : 1 ≤ p.natDegree) (f : R →+* S) (x : S) : Polynomial.eval₂ f (f p.leadingCoeff * x) (normalizeScaleRoots p) = f p.leadingCoeff ^ (p.natDegree - 1) * Polynomial.eval₂ f x p

theorem normalizeScaleRoots_monic {R : Type u_1} [CommRing R] (p : Polynomial R) (h : p ≠ 0) : (normalizeScaleRoots p).Monic

theorem RingHom.isIntegralElem_leadingCoeff_mul {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (p : Polynomial R) (x : S) (h : Polynomial.eval₂ f x p = 0) : f.IsIntegralElem (f p.leadingCoeff * x)

Given a p : R[X] and a x : S such that p.eval₂ f x = 0, f p.leadingCoeff * x is integral.

theorem isIntegral_leadingCoeff_smul {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (p : Polynomial R) (x : S) [Algebra R S] (h : (Polynomial.aeval x) p = 0) : IsIntegral R (p.leadingCoeff • x)

Given a p : R[X] and a root x : S, then p.leadingCoeff • x : S is integral over R.

class IsIntegralClosure (A : Type u_1) (R : Type u_2) (B : Type u_3) [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B] [Algebra A B] : Prop

IsIntegralClosure A R B is the characteristic predicate stating A is the integral closure of R in B, i.e. that an element of B is integral over R iff it is an element of (the image of) A.

• algebraMap_injective' : Function.Injective ⇑(algebraMap A B)
• isIntegral_iff : ∀ {x : B}, IsIntegral R x ↔ ∃ (y : A), (algebraMap A B) y = x

theorem IsIntegralClosure.algebraMap_injective' {A : Type u_1} (R : Type u_2) {B : Type u_3} [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B] [Algebra A B] [self : IsIntegralClosure A R B] : Function.Injective ⇑(algebraMap A B)

theorem IsIntegralClosure.isIntegral_iff {A : Type u_1} {R : Type u_2} {B : Type u_3} [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B] [Algebra A B] [self : IsIntegralClosure A R B] {x : B} : IsIntegral R x ↔ ∃ (y : A), (algebraMap A B) y = x

instance integralClosure.isIntegralClosure (R : Type u_1) (A : Type u_2) [CommRing R] [CommRing A] [Algebra R A] : IsIntegralClosure (↥(integralClosure R A)) R A

Equations

• ⋯ = ⋯

theorem IsIntegralClosure.algebraMap_injective (A : Type u_1) (R : Type u_2) (B : Type u_3) [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] : Function.Injective ⇑(algebraMap A B)

theorem IsIntegralClosure.isIntegral (R : Type u_1) {A : Type u_2} (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] [Algebra R A] [IsScalarTower R A B] (x : A) : IsIntegral R x

theorem IsIntegralClosure.isIntegral_algebra (R : Type u_1) {A : Type u_2} (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] [Algebra R A] [IsScalarTower R A B] : Algebra.IsIntegral R A

theorem IsIntegralClosure.noZeroSMulDivisors (R : Type u_1) {A : Type u_2} (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] [Algebra R A] [IsScalarTower R A B] [NoZeroSMulDivisors R B] : NoZeroSMulDivisors R A

noncomputable def IsIntegralClosure.mk' {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (x : B) (hx : IsIntegral R x) : A

If x : B is integral over R, then it is an element of the integral closure of R in B.

Equations

• IsIntegralClosure.mk' A x hx = Classical.choose ⋯

@[simp]

theorem IsIntegralClosure.algebraMap_mk' {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (x : B) (hx : IsIntegral R x) : (algebraMap A B) (IsIntegralClosure.mk' A x hx) = x

@[simp]

theorem IsIntegralClosure.mk'_one {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (h : optParam (IsIntegral R 1) ⋯) : IsIntegralClosure.mk' A 1 h = 1

@[simp]

theorem IsIntegralClosure.mk'_zero {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (h : optParam (IsIntegral R 0) ⋯) : IsIntegralClosure.mk' A 0 h = 0

theorem IsIntegralClosure.mk'_add {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (x : B) (y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) : IsIntegralClosure.mk' A (x + y) ⋯ = IsIntegralClosure.mk' A x hx + IsIntegralClosure.mk' A y hy

theorem IsIntegralClosure.mk'_mul {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (x : B) (y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) : IsIntegralClosure.mk' A (x * y) ⋯ = IsIntegralClosure.mk' A x hx * IsIntegralClosure.mk' A y hy

@[simp]

theorem IsIntegralClosure.mk'_algebraMap {R : Type u_1} (A : Type u_2) {B : Type u_3} [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] [Algebra R A] [IsScalarTower R A B] (x : R) (h : optParam (IsIntegral R ((algebraMap R B) x)) ⋯) : IsIntegralClosure.mk' A ((algebraMap R B) x) h = (algebraMap R A) x

noncomputable def IsIntegralClosure.lift {R : Type u_1} (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] {S : Type u_4} [CommRing S] [Algebra R S] [Algebra S B] [IsScalarTower R S B] [Algebra R A] [IsScalarTower R A B] (h : Algebra.IsIntegral R S) : S →ₐ[R] A

If B / S / R is a tower of ring extensions where S is integral over R, then S maps (uniquely) into an integral closure B / A / R.

Equations

• IsIntegralClosure.lift A B h = { toFun := fun (x : S) => IsIntegralClosure.mk' A ((algebraMap S B) x) ⋯, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯, commutes' := ⋯ }

@[simp]

theorem IsIntegralClosure.algebraMap_lift {R : Type u_1} (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] {S : Type u_4} [CommRing S] [Algebra R S] [Algebra S B] [IsScalarTower R S B] [Algebra R A] [IsScalarTower R A B] (h : Algebra.IsIntegral R S) (x : S) : (algebraMap A B) ((IsIntegralClosure.lift A B h) x) = (algebraMap S B) x

noncomputable def IsIntegralClosure.equiv (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (A' : Type u_4) [CommRing A'] [Algebra A' B] [IsIntegralClosure A' R B] [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A' B] : A ≃ₐ[R] A'

Integral closures are all isomorphic to each other.

Equations

• IsIntegralClosure.equiv R A B A' = AlgEquiv.ofAlgHom (IsIntegralClosure.lift A' B ⋯) (IsIntegralClosure.lift A B ⋯) ⋯ ⋯

@[simp]

theorem IsIntegralClosure.algebraMap_equiv (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommRing R] [CommRing A] [CommRing B] [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] (A' : Type u_4) [CommRing A'] [Algebra A' B] [IsIntegralClosure A' R B] [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A' B] (x : A) : (algebraMap A' B) ((IsIntegralClosure.equiv R A B A') x) = (algebraMap A B) x

theorem isIntegral_trans {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra A B] [Algebra R B] [Algebra R A] [IsScalarTower R A B] (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegral A x) : IsIntegral R x

If A is an R-algebra all of whose elements are integral over R, and x is an element of an A-algebra that is integral over A, then x is integral over R.

theorem Algebra.IsIntegral.trans {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra A B] [Algebra R B] [Algebra R A] [IsScalarTower R A B] (hA : Algebra.IsIntegral R A) (hB : Algebra.IsIntegral A B) : Algebra.IsIntegral R B

If A is an R-algebra all of whose elements are integral over R, and B is an A-algebra all of whose elements are integral over A, then all elements of B are integral over R.

theorem RingHom.IsIntegral.trans {R : Type u_1} {S : Type u_4} {T : Type u_5} [CommRing R] [CommRing S] [CommRing T] (f : R →+* S) (g : S →+* T) (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.comp f).IsIntegral

theorem IsIntegralClosure.tower_top {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {B : Type u_6} {C : Type u_7} [CommRing C] [CommRing B] [Algebra R B] [Algebra A B] [Algebra C B] [IsScalarTower R A B] [IsIntegralClosure C R B] (hRA : Algebra.IsIntegral R A) : IsIntegralClosure C A B

If R → A → B is an algebra tower, C is the integral closure of R in B and A is integral over R, then C is the integral closure of A in B.

theorem RingHom.isIntegral_of_surjective {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) : f.IsIntegral

theorem Algebra.isIntegral_of_surjective {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] (h : Function.Surjective ⇑(algebraMap R A)) : Algebra.IsIntegral R A

theorem IsIntegral.tower_bot {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommRing R] [CommRing A] [Ring B] [Algebra A B] [Algebra R B] [Algebra R A] [IsScalarTower R A B] (H : Function.Injective ⇑(algebraMap A B)) {x : A} (h : IsIntegral R ((algebraMap A B) x)) : IsIntegral R x

If R → A → B is an algebra tower with A → B injective, then if the entire tower is an integral extension so is R → A

theorem RingHom.IsIntegral.tower_bot {R : Type u_1} {S : Type u_4} {T : Type u_5} [CommRing R] [CommRing S] [CommRing T] (f : R →+* S) (g : S →+* T) (hg : Function.Injective ⇑g) (hfg : (g.comp f).IsIntegral) : f.IsIntegral

theorem IsIntegral.tower_bot_of_field {R : Type u_6} {A : Type u_7} {B : Type u_8} [CommRing R] [Field A] [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] {x : A} (h : IsIntegral R ((algebraMap A B) x)) : IsIntegral R x

theorem RingHom.isIntegralElem.of_comp {R : Type u_1} {S : Type u_4} {T : Type u_5} [CommRing R] [CommRing S] [CommRing T] (f : R →+* S) (g : S →+* T) {x : T} (h : (g.comp f).IsIntegralElem x) : g.IsIntegralElem x

theorem RingHom.IsIntegral.tower_top {R : Type u_1} {S : Type u_4} {T : Type u_5} [CommRing R] [CommRing S] [CommRing T] (f : R →+* S) (g : S →+* T) (h : (g.comp f).IsIntegral) : g.IsIntegral

theorem RingHom.IsIntegral.quotient {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {I : Ideal S} (hf : f.IsIntegral) : (Ideal.quotientMap I f ⋯).IsIntegral

theorem Algebra.IsIntegral.quotient {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] {I : Ideal A} (hRA : Algebra.IsIntegral R A) : Algebra.IsIntegral (R ⧸ Ideal.comap (algebraMap R A) I) (A ⧸ I)

theorem isIntegral_quotientMap_iff {R : Type u_1} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) {I : Ideal S} : (Ideal.quotientMap I f ⋯).IsIntegral ↔ ((Ideal.Quotient.mk I).comp f).IsIntegral

theorem isField_of_isIntegral_of_isField {R : Type u_6} {S : Type u_7} [CommRing R] [CommRing S] [Algebra R S] (H : Algebra.IsIntegral R S) (hRS : Function.Injective ⇑(algebraMap R S)) (hS : IsField S) : IsField R

If the integral extension R → S is injective, and S is a field, then R is also a field.

theorem isField_of_isIntegral_of_isField' {R : Type u_6} {S : Type u_7} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S) (hR : IsField R) : IsField S

theorem Algebra.IsIntegral.isField_iff_isField {R : Type u_6} {S : Type u_7} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S) (hRS : Function.Injective ⇑(algebraMap R S)) : IsField R ↔ IsField S

theorem integralClosure_idem {R : Type u_1} {A : Type u_2} [CommRing R] [CommRing A] [Algebra R A] : integralClosure (↑↑(integralClosure R A)) A = ⊥

instance instIsDomainSubtypeMemSubalgebraIntegralClosure {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S] : IsDomain ↥(integralClosure R S)

Equations

• ⋯ = ⋯

theorem roots_mem_integralClosure {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S] {f : Polynomial R} (hf : f.Monic) {a : S} (ha : a ∈ f.aroots S) : a ∈ integralClosure R S