Loogle!
Result
Found 38 definitions mentioning Algebra.adjoin and Polynomial.
- Algebra.adjoin_singleton_eq_range_aeval Mathlib.Algebra.Polynomial.AlgebraMap
∀ (R : Type u) {A : Type z} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] (x : A), Algebra.adjoin R {x} = (Polynomial.aeval x).range - Polynomial.aeval_mem_adjoin_singleton Mathlib.Algebra.Polynomial.AlgebraMap
∀ (R : Type u) {A : Type z} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A] {p : Polynomial R} (x : A), (Polynomial.aeval x) p ∈ Algebra.adjoin R {x} - Polynomial.adjoin_X Mathlib.Algebra.Polynomial.AlgebraMap
∀ {R : Type u} [inst : CommSemiring R], Algebra.adjoin R {Polynomial.X} = ⊤ - Submodule.span_range_natDegree_eq_adjoin Mathlib.RingTheory.IntegralClosure
∀ {R : Type u_5} {A : Type u_6} [inst : CommRing R] [inst_1 : Semiring A] [inst_2 : Algebra R A] {x : A} {f : Polynomial R}, f.Monic → (Polynomial.aeval x) f = 0 → Submodule.span R ↑(Finset.image (fun x_1 => x ^ x_1) (Finset.range f.natDegree)) = Subalgebra.toSubmodule (Algebra.adjoin R {x}) - Polynomial.adjoin_rootSet_eq_range Mathlib.Algebra.Polynomial.Splits
∀ {R : Type u_1} {K : Type v} {L : Type w} [inst : CommRing R] [inst_1 : Field K] [inst_2 : Field L] [inst_3 : Algebra R K] [inst_4 : Algebra R L] {p : Polynomial R}, Polynomial.Splits (algebraMap R K) p → ∀ (f : K →ₐ[R] L), Algebra.adjoin R (p.rootSet L) = f.range ↔ Algebra.adjoin R (p.rootSet K) = ⊤ - Polynomial.IsWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegree Mathlib.RingTheory.Polynomial.Eisenstein.Basic
∀ {R : Type u} [inst : CommRing R] {f : Polynomial R} {S : Type v} [inst_1 : CommRing S] [inst_2 : Algebra R S] {p : R} {x : S}, (Polynomial.aeval x) f = 0 → f.Monic → f.IsWeaklyEisensteinAt (Submodule.span R {p}) → ∃ y ∈ Algebra.adjoin R {x}, (algebraMap R S) p * y = x ^ (Polynomial.map (algebraMap R S) f).natDegree - Polynomial.IsWeaklyEisensteinAt.exists_mem_adjoin_mul_eq_pow_natDegree_le Mathlib.RingTheory.Polynomial.Eisenstein.Basic
∀ {R : Type u} [inst : CommRing R] {f : Polynomial R} {S : Type v} [inst_1 : CommRing S] [inst_2 : Algebra R S] {p : R} {x : S}, (Polynomial.aeval x) f = 0 → f.Monic → f.IsWeaklyEisensteinAt (Submodule.span R {p}) → ∀ (i : ℕ), (Polynomial.map (algebraMap R S) f).natDegree ≤ i → ∃ y ∈ Algebra.adjoin R {x}, (algebraMap R S) p * y = x ^ i - AdjoinRoot.adjoinRoot_eq_top Mathlib.RingTheory.AdjoinRoot
∀ {R : Type u} [inst : CommRing R] {f : Polynomial R}, Algebra.adjoin R {AdjoinRoot.root f} = ⊤ - AdjoinRoot.Minpoly.toAdjoin.apply_X Mathlib.RingTheory.AdjoinRoot
∀ {R : Type u} {S : Type v} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {x : S}, (AdjoinRoot.Minpoly.toAdjoin R x) ((AdjoinRoot.mk (minpoly R x)) Polynomial.X) = ⟨x, ⋯⟩ - AdjoinRoot.Minpoly.toAdjoin_apply Mathlib.RingTheory.AdjoinRoot
∀ (R : Type u) {S : Type v} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (x : S) (a : Polynomial R ⧸ Submodule.toAddSubgroup (Ideal.span {minpoly R x})), (AdjoinRoot.Minpoly.toAdjoin R x) a = (QuotientAddGroup.lift (Submodule.toAddSubgroup (Ideal.span {minpoly R x})) ↑(Polynomial.eval₂RingHom (algebraMap R ↥(Algebra.adjoin R {x})) ⟨x, ⋯⟩) ⋯) a - AdjoinRoot.Minpoly.toAdjoin_apply' Mathlib.RingTheory.AdjoinRoot
∀ {R : Type u} {S : Type v} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {x : S} (a : AdjoinRoot (minpoly R x)), (AdjoinRoot.Minpoly.toAdjoin R x) a = (AdjoinRoot.liftHom (minpoly R x) ⟨x, ⋯⟩ ⋯) a - Algebra.adjoin.liftSingleton Mathlib.RingTheory.Adjoin.Field
(F : Type u_1) → [inst : Field F] → {S : Type u_2} → {T : Type u_3} → [inst_1 : CommRing S] → [inst_2 : CommRing T] → [inst_3 : Algebra F S] → [inst_4 : Algebra F T] → (x : S) → (y : T) → (Polynomial.aeval y) (minpoly F x) = 0 → ↥(Algebra.adjoin F {x}) →ₐ[F] T - Polynomial.IsSplittingField.adjoin_rootSet Mathlib.FieldTheory.SplittingField.IsSplittingField
∀ {K : Type v} (L : Type w) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (f : Polynomial K) [inst_3 : Polynomial.IsSplittingField K L f], Algebra.adjoin K (f.rootSet L) = ⊤ - Polynomial.IsSplittingField.adjoin_rootSet' Mathlib.FieldTheory.SplittingField.IsSplittingField
∀ {K : Type v} {L : Type w} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] {f : Polynomial K} [self : Polynomial.IsSplittingField K L f], Algebra.adjoin K (f.rootSet L) = ⊤ - Polynomial.IsSplittingField.adjoin_rootSet_eq_range Mathlib.FieldTheory.SplittingField.IsSplittingField
∀ {F : Type u} {K : Type v} (L : Type w) [inst : Field K] [inst_1 : Field L] [inst_2 : Field F] [inst_3 : Algebra K L] [inst_4 : Algebra K F] (f : Polynomial K) [inst_5 : Polynomial.IsSplittingField K L f] (i : L →ₐ[K] F), Algebra.adjoin K (f.rootSet F) = i.range - Polynomial.IsSplittingField.mk Mathlib.FieldTheory.SplittingField.IsSplittingField
∀ {K : Type v} {L : Type w} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] {f : Polynomial K}, Polynomial.Splits (algebraMap K L) f → Algebra.adjoin K (f.rootSet L) = ⊤ → Polynomial.IsSplittingField K L f - Polynomial.SplittingFieldAux.adjoin_rootSet Mathlib.FieldTheory.SplittingField.Construction
∀ (n : ℕ) {K : Type u} [inst : Field K] (f : Polynomial K), f.natDegree = n → Algebra.adjoin K (f.rootSet (Polynomial.SplittingFieldAux n f)) = ⊤ - Polynomial.SplittingField.adjoin_rootSet Mathlib.FieldTheory.SplittingField.Construction
∀ {K : Type v} [inst : Field K] (f : Polynomial K), Algebra.adjoin K (f.rootSet f.SplittingField) = ⊤ - Module.End.exists_isNilpotent_isSemisimple_of_separable_of_dvd_pow Mathlib.LinearAlgebra.JordanChevalley
∀ {K : Type u_1} {V : Type u_2} [inst : Field K] [inst_1 : AddCommGroup V] [inst_2 : Module K V] {f : Module.End K V} {P : Polynomial K} {k : ℕ}, P.Separable → minpoly K f ∣ P ^ k → ∃ n ∈ Algebra.adjoin K {f}, ∃ s ∈ Algebra.adjoin K {f}, IsNilpotent n ∧ s.IsSemisimple ∧ f = n + s - polynomialFunctions.eq_adjoin_X Mathlib.Topology.ContinuousFunction.Polynomial
∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : TopologicalSpace R] [inst_2 : TopologicalSemiring R] (s : Set R), polynomialFunctions s = Algebra.adjoin R {(Polynomial.toContinuousMapOnAlgHom s) Polynomial.X} - minpoly.equivAdjoin_symm_apply Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : IsDomain R] [inst_3 : Algebra R S] [inst_4 : IsDomain S] [inst_5 : NoZeroSMulDivisors R S] [inst_6 : IsIntegrallyClosed R] {x : S} (hx : IsIntegral R x) (b : ↥(Algebra.adjoin R {x})), (minpoly.equivAdjoin hx).symm b = Function.surjInv ⋯ b - minpoly.equivAdjoin_apply Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : IsDomain R] [inst_3 : Algebra R S] [inst_4 : IsDomain S] [inst_5 : NoZeroSMulDivisors R S] [inst_6 : IsIntegrallyClosed R] {x : S} (hx : IsIntegral R x) (a : AdjoinRoot (minpoly R x)), (minpoly.equivAdjoin hx) a = (QuotientAddGroup.lift (Submodule.toAddSubgroup (Ideal.span {minpoly R x})) ↑(Polynomial.eval₂RingHom (algebraMap R ↥(Algebra.adjoin R {x})) ⟨x, ⋯⟩) ⋯) a - sum_smul_minpolyDiv_eq_X_pow Mathlib.FieldTheory.Minpoly.MinpolyDiv
∀ {K : Type u_2} {L : Type u_3} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (E : Type u_1) [inst_3 : Field E] [inst_4 : Algebra K E] [inst_5 : IsAlgClosed E] [inst_6 : FiniteDimensional K L] [inst_7 : IsSeparable K L] {x : L}, Algebra.adjoin K {x} = ⊤ → ∀ {r : ℕ}, r < FiniteDimensional.finrank K L → (Finset.univ.sum fun σ => Polynomial.map (↑σ) ((x ^ r / (Polynomial.aeval x) (Polynomial.derivative (minpoly K x))) • minpolyDiv K x)) = Polynomial.X ^ r - AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin Mathlib.RingTheory.AlgebraicIndependent
{ι : Type u_1} → {R : Type u_3} → {A : Type u_5} → {x : ι → A} → [inst : CommRing R] → [inst_1 : CommRing A] → [inst_2 : Algebra R A] → AlgebraicIndependent R x → MvPolynomial (Option ι) R ≃+* Polynomial ↥(Algebra.adjoin R (Set.range x)) - AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_none Mathlib.RingTheory.AlgebraicIndependent
∀ {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] (hx : AlgebraicIndependent R x), hx.mvPolynomialOptionEquivPolynomialAdjoin (MvPolynomial.X none) = Polynomial.X - AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_some Mathlib.RingTheory.AlgebraicIndependent
∀ {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] (hx : AlgebraicIndependent R x) (i : ι), hx.mvPolynomialOptionEquivPolynomialAdjoin (MvPolynomial.X (some i)) = Polynomial.C (hx.aevalEquiv (MvPolynomial.X i)) - AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_C Mathlib.RingTheory.AlgebraicIndependent
∀ {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] (hx : AlgebraicIndependent R x) (r : R), hx.mvPolynomialOptionEquivPolynomialAdjoin (MvPolynomial.C r) = Polynomial.C ((algebraMap R ↥(Algebra.adjoin R (Set.range x))) r) - AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_apply Mathlib.RingTheory.AlgebraicIndependent
∀ {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] (hx : AlgebraicIndependent R x) (y : MvPolynomial (Option ι) R), hx.mvPolynomialOptionEquivPolynomialAdjoin y = Polynomial.map (↑hx.aevalEquiv) ((MvPolynomial.aeval fun o => o.elim Polynomial.X fun s => Polynomial.C (MvPolynomial.X s)) y) - AlgebraicIndependent.aeval_comp_mvPolynomialOptionEquivPolynomialAdjoin Mathlib.RingTheory.AlgebraicIndependent
∀ {ι : Type u_1} {R : Type u_3} {A : Type u_5} {x : ι → A} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] (hx : AlgebraicIndependent R x) (a : A), (↑(Polynomial.aeval a)).comp hx.mvPolynomialOptionEquivPolynomialAdjoin.toRingHom = ↑(MvPolynomial.aeval fun o => o.elim a x) - Algebra.adjoin_root_eq_top_of_isSplittingField Mathlib.FieldTheory.KummerExtension
∀ {K : Type u} [inst : Field K] {n : ℕ}, (primitiveRoots n K).Nonempty → ∀ {a : K}, Irreducible (Polynomial.X ^ n - Polynomial.C a) → ∀ {L : Type u_1} [inst_1 : Field L] [inst_2 : Algebra K L] [inst_3 : Polynomial.IsSplittingField K L (Polynomial.X ^ n - Polynomial.C a)] {α : L}, α ^ n = (algebraMap K L) a → Algebra.adjoin K {α} = ⊤ - adjoin_monomial_eq_reesAlgebra Mathlib.RingTheory.ReesAlgebra
∀ {R : Type u} [inst : CommRing R] (I : Ideal R), Algebra.adjoin R ↑(Submodule.map (Polynomial.monomial 1) I) = reesAlgebra I - monomial_mem_adjoin_monomial Mathlib.RingTheory.ReesAlgebra
∀ {R : Type u} [inst : CommRing R] {I : Ideal R} {n : ℕ} {r : R}, r ∈ I ^ n → (Polynomial.monomial n) r ∈ Algebra.adjoin R ↑(Submodule.map (Polynomial.monomial 1) I) - mem_adjoin_of_dvd_coeff_of_dvd_aeval Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
∀ {A : Type u_1} {B : Type u_2} [inst : CommSemiring A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : NoZeroSMulDivisors A B] {Q : Polynomial A} {p : A} {x z : B}, p ≠ 0 → (∀ i ∈ Finset.range (Q.natDegree + 1), p ∣ Q.coeff i) → (Polynomial.aeval x) Q = p • z → z ∈ Algebra.adjoin A {x} - Algebra.adjoin.powerBasis'_minpoly_gen Mathlib.RingTheory.IsAdjoinRoot
∀ {R : Type u} {S : Type v} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [inst_3 : IsDomain R] [inst_4 : IsDomain S] [inst_5 : NoZeroSMulDivisors R S] [inst_6 : IsIntegrallyClosed R] {x : S} (hx' : IsIntegral R x), minpoly R x = minpoly R (Algebra.adjoin.powerBasis' hx').gen - KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk.eq_1 Mathlib.NumberTheory.KummerDedekind
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {x : S} {I : Ideal R} [inst_3 : IsDomain R] [inst_4 : IsIntegrallyClosed R] [inst_5 : IsDedekindDomain S] [inst_6 : NoZeroSMulDivisors R S] (hI : I.IsMaximal) (hI' : I ≠ ⊥) (hx : Ideal.comap (algebraMap R S) (conductor R x) ⊔ I = ⊤) (hx' : IsIntegral R x), KummerDedekind.normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx' = let_fun this := ⋯; let f := (quotAdjoinEquivQuotMap hx ⋯).symm.trans (((Algebra.adjoin.powerBasis' hx').quotientEquivQuotientMinpolyMap I).toRingEquiv.trans (Ideal.quotEquivOfEq ⋯)); (normalizedFactorsEquivOfQuotEquiv f ⋯ ⋯).trans (normalizedFactorsEquivSpanNormalizedFactors ⋯).symm - aeval_derivative_mem_differentIdeal Mathlib.RingTheory.DedekindDomain.Different
∀ (A : Type u_1) (K : Type u_2) (L : Type u) {B : Type u_3} [inst : CommRing A] [inst_1 : Field K] [inst_2 : CommRing B] [inst_3 : Field L] [inst_4 : Algebra A K] [inst_5 : Algebra B L] [inst_6 : Algebra A B] [inst_7 : Algebra K L] [inst_8 : Algebra A L] [inst_9 : IsScalarTower A K L] [inst_10 : IsScalarTower A B L] [inst_11 : IsDomain A] [inst_12 : IsDomain B] [inst_13 : IsFractionRing A K] [inst_14 : IsIntegralClosure B A L] [inst_15 : FiniteDimensional K L] [inst_16 : IsSeparable K L] [inst_17 : IsIntegrallyClosed A] [inst_18 : IsDedekindDomain B] [inst_19 : NoZeroSMulDivisors A B] (x : B), Algebra.adjoin K {(algebraMap B L) x} = ⊤ → (Polynomial.aeval x) (Polynomial.derivative (minpoly A x)) ∈ differentIdeal A B - conductor_mul_differentIdeal Mathlib.RingTheory.DedekindDomain.Different
∀ (A : Type u_1) (K : Type u_2) (L : Type u) {B : Type u_3} [inst : CommRing A] [inst_1 : Field K] [inst_2 : CommRing B] [inst_3 : Field L] [inst_4 : Algebra A K] [inst_5 : Algebra B L] [inst_6 : Algebra A B] [inst_7 : Algebra K L] [inst_8 : Algebra A L] [inst_9 : IsScalarTower A K L] [inst_10 : IsScalarTower A B L] [inst_11 : IsDomain A] [inst_12 : IsDomain B] [inst_13 : IsFractionRing A K] [inst_14 : IsIntegralClosure B A L] [inst_15 : FiniteDimensional K L] [inst_16 : IsSeparable K L] [inst_17 : IsIntegrallyClosed A] [inst_18 : IsDedekindDomain B] [inst_19 : NoZeroSMulDivisors A B] (x : B), Algebra.adjoin K {(algebraMap B L) x} = ⊤ → conductor A x * differentIdeal A B = Ideal.span {(Polynomial.aeval x) (Polynomial.derivative (minpoly A x))} - traceForm_dualSubmodule_adjoin Mathlib.RingTheory.DedekindDomain.Different
∀ (A : Type u_1) (K : Type u_2) {L : Type u} [inst : CommRing A] [inst_1 : Field K] [inst_2 : Field L] [inst_3 : Algebra A K] [inst_4 : Algebra K L] [inst_5 : Algebra A L] [inst_6 : IsScalarTower A K L] [inst_7 : IsDomain A] [inst_8 : IsFractionRing A K] [inst_9 : FiniteDimensional K L] [inst_10 : IsSeparable K L] [inst_11 : IsIntegrallyClosed A] {x : L}, Algebra.adjoin K {x} = ⊤ → IsIntegral A x → (Algebra.traceForm K L).dualSubmodule (Subalgebra.toSubmodule (Algebra.adjoin A {x})) = ((Polynomial.aeval x) (Polynomial.derivative (minpoly K x)))⁻¹ • Subalgebra.toSubmodule (Algebra.adjoin A {x})
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About
Loogle searches of Lean and Mathlib definitions and theorems.
You may also want to try the CLI version, the VS
Code extension, the lean.nvim
integration or the Zulip bot.
Usage
Loogle finds definitions and lemmas in various ways:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.By subexpression:
🔍_ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.The pattern can also be non-linear, as in
🔍Real.sqrt ?a * Real.sqrt ?a
If the pattern has parameters, they are matched in any order. Both of these will find
List.map
:
🔍(?a -> ?b) -> List ?a -> List ?b
🔍List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍|- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all→
and∀
) has the given shape.As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍|- _ < _ → tsum _ < tsum _
will findtsum_lt_tsum
even though the hypothesisf i < g i
is not the last.
If you pass more than one such search filter, separated by commas
Loogle will return lemmas which match all of them. The
search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
woould find all lemmas which mention the constants Real.sin
and tsum
, have "two"
as a substring of the
lemma name, include a product and a power somewhere in the type,
and have a hypothesis of the form _ < _
(if
there were any such lemmas). Metavariables (?a
) are
assigned independently in each filter.
The #lucky
button will directly send you to the
documentation of the first hit.
Source code
You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is currently provided by Joachim Breitner <mail@joachim-breitner.de>.
This is Loogle revision fa2ddf5
serving mathlib revision 38aa2fc