statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
polynomial.quot_quot_equiv_comm :
(R ⧸ I)[X] ⧸ span ({f.map (I^.quotient.mk)} : set (polynomial (R ⧸ I))) ≃+*
(R[X] ⧸ map C I) ⧸ span ({(ideal.quotient.mk (I.map C)) f} : set (R[X] ⧸ map C I)) | quotient_equiv (span ({f.map (I^.quotient.mk)} : set (polynomial (R ⧸ I))))
(span {ideal.quotient.mk (I.map polynomial.C) f})
(polynomial_quotient_equiv_quotient_polynomial I)
(by rw [map_span, set.image_singleton, ring_equiv.coe_to_ring_hom,
polynomial_quotient_equiv_quotient_polynomial_map_mk I f]) | def | adjoin_root.polynomial.quot_quot_equiv_comm | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"ideal.quotient.mk",
"polynomial",
"polynomial.C",
"ring_equiv.coe_to_ring_hom",
"set.image_singleton"
] | The natural isomorphism `(R/I)[x]/(f mod I) ≅ (R[x]/I*R[x])/(f mod I[x])` where
`f : R[X]` and `I : ideal R` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial.quot_quot_equiv_comm_mk (p : R[X]) :
(polynomial.quot_quot_equiv_comm I f) (ideal.quotient.mk _ (p.map I^.quotient.mk)) =
(ideal.quotient.mk _ (ideal.quotient.mk _ p)) | by simp only [polynomial.quot_quot_equiv_comm, quotient_equiv_mk,
polynomial_quotient_equiv_quotient_polynomial_map_mk] | lemma | adjoin_root.polynomial.quot_quot_equiv_comm_mk | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial.quot_quot_equiv_comm_symm_mk_mk (p : R[X]) :
(polynomial.quot_quot_equiv_comm I f).symm (ideal.quotient.mk _ (ideal.quotient.mk _ p)) =
(ideal.quotient.mk _ (p.map I^.quotient.mk)) | by simp only [polynomial.quot_quot_equiv_comm, quotient_equiv_symm_mk,
polynomial_quotient_equiv_quotient_polynomial_symm_mk] | lemma | adjoin_root.polynomial.quot_quot_equiv_comm_symm_mk_mk | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_adjoin_root_equiv_quot_polynomial_quot : (adjoin_root f) ⧸ (I.map (of f)) ≃+*
(R ⧸ I)[X] ⧸ (span ({f.map (I^.quotient.mk)} : set (R ⧸ I)[X])) | (quot_map_of_equiv_quot_map_C_map_span_mk I f).trans
((quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk I f).trans
((ideal.quot_equiv_of_eq
(show (span ({f} : set R[X])).map (I.map (C : R →+* R[X]))^.quotient.mk =
span ({(ideal.quotient.mk (I.map polynomial.C)) f} : set (R[X] ⧸ map C I)),
from by ... | def | adjoin_root.quot_adjoin_root_equiv_quot_polynomial_quot | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"adjoin_root",
"ideal.quot_equiv_of_eq",
"ideal.quotient.mk",
"polynomial.C",
"set.image_singleton"
] | The natural isomorphism `R[α]/I[α] ≅ (R/I)[X]/(f mod I)` for `α` a root of `f : R[X]`
and `I : ideal R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_adjoin_root_equiv_quot_polynomial_quot_mk_of (p : R[X]) :
quot_adjoin_root_equiv_quot_polynomial_quot I f (ideal.quotient.mk (I.map (of f)) (mk f p)) =
ideal.quotient.mk (span ({f.map (I^.quotient.mk)} : set (R ⧸ I)[X]))
(p.map I^.quotient.mk) | by rw [quot_adjoin_root_equiv_quot_polynomial_quot, ring_equiv.trans_apply, ring_equiv.trans_apply,
ring_equiv.trans_apply, quot_map_of_equiv_quot_map_C_map_span_mk_mk,
quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk_mk, quot_quot_mk, ring_hom.comp_apply,
quot_equiv_of_eq_mk, polynomial.quot_quot_e... | lemma | adjoin_root.quot_adjoin_root_equiv_quot_polynomial_quot_mk_of | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"ideal.quotient.mk",
"ring_equiv.trans_apply",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_adjoin_root_equiv_quot_polynomial_quot_symm_mk_mk (p : R[X]) :
(quot_adjoin_root_equiv_quot_polynomial_quot I f).symm
(ideal.quotient.mk (span ({f.map (I^.quotient.mk)} : set (R ⧸ I)[X]))
(p.map I^.quotient.mk)) = (ideal.quotient.mk (I.map (of f)) (mk f p)) | by rw [quot_adjoin_root_equiv_quot_polynomial_quot, ring_equiv.symm_trans_apply,
ring_equiv.symm_trans_apply, ring_equiv.symm_trans_apply, ring_equiv.symm_symm,
polynomial.quot_quot_equiv_comm_mk, ideal.quot_equiv_of_eq_symm,
ideal.quot_equiv_of_eq_mk, ← ring_hom.comp_apply, ← double_quot.quot_quot_mk,
... | lemma | adjoin_root.quot_adjoin_root_equiv_quot_polynomial_quot_symm_mk_mk | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"double_quot.quot_quot_mk",
"ideal.quot_equiv_of_eq_mk",
"ideal.quot_equiv_of_eq_symm",
"ideal.quotient.mk",
"ring_equiv.symm_symm",
"ring_equiv.symm_trans_apply",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_equiv_quot_map (f : R[X]) (I : ideal R) :
((adjoin_root f) ⧸ (ideal.map (of f) I)) ≃ₐ[R]
((R ⧸ I) [X]) ⧸ (ideal.span ({polynomial.map I^.quotient.mk f} : set ((R ⧸ I) [X]))) | alg_equiv.of_ring_equiv (show ∀ x, (quot_adjoin_root_equiv_quot_polynomial_quot I f)
(algebra_map R _ x) = algebra_map R _ x, from λ x, begin
have : algebra_map R ((adjoin_root f) ⧸ (ideal.map (of f) I)) x = ideal.quotient.mk
(ideal.map (adjoin_root.of f) I) ((mk f) (C x)) := rfl,
simpa only [this, quot... | def | adjoin_root.quot_equiv_quot_map | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"adjoin_root",
"adjoin_root.of",
"alg_equiv.of_ring_equiv",
"algebra_map",
"ideal",
"ideal.map",
"ideal.quotient.mk",
"ideal.span",
"polynomial.map"
] | Promote `adjoin_root.quot_adjoin_root_equiv_quot_polynomial_quot` to an alg_equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_equiv_quot_map_apply_mk (f g : R[X]) (I : ideal R) :
adjoin_root.quot_equiv_quot_map f I (ideal.quotient.mk _ (adjoin_root.mk f g)) =
ideal.quotient.mk _ (g.map I^.quotient.mk) | by rw [adjoin_root.quot_equiv_quot_map_apply,
adjoin_root.quot_adjoin_root_equiv_quot_polynomial_quot_mk_of] | lemma | adjoin_root.quot_equiv_quot_map_apply_mk | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"adjoin_root.mk",
"adjoin_root.quot_adjoin_root_equiv_quot_polynomial_quot_mk_of",
"adjoin_root.quot_equiv_quot_map",
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_equiv_quot_map_symm_apply_mk (f g : R[X]) (I : ideal R) :
(adjoin_root.quot_equiv_quot_map f I).symm (ideal.quotient.mk _ (map (ideal.quotient.mk I) g)) =
ideal.quotient.mk _ (adjoin_root.mk f g) | by rw [adjoin_root.quot_equiv_quot_map_symm_apply,
adjoin_root.quot_adjoin_root_equiv_quot_polynomial_quot_symm_mk_mk] | lemma | adjoin_root.quot_equiv_quot_map_symm_apply_mk | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"adjoin_root.mk",
"adjoin_root.quot_adjoin_root_equiv_quot_polynomial_quot_symm_mk_mk",
"adjoin_root.quot_equiv_quot_map",
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_equiv_quotient_minpoly_map (pb : power_basis R S)
(I : ideal R) :
(S ⧸ I.map (algebra_map R S)) ≃ₐ[R] (polynomial (R ⧸ I)) ⧸
(ideal.span ({(minpoly R pb.gen).map I^.quotient.mk} : set (polynomial (R ⧸ I)))) | (of_ring_equiv
(show ∀ x, (ideal.quotient_equiv _ (ideal.map (adjoin_root.of (minpoly R pb.gen)) I)
(adjoin_root.equiv' (minpoly R pb.gen) pb
(by rw [adjoin_root.aeval_eq, adjoin_root.mk_self])
(minpoly.aeval _ _)).symm.to_ring_equiv
(by rw [ideal.map_map, alg_equiv.to_ring_equiv_eq_coe, ← alg_equiv.c... | def | power_basis.quotient_equiv_quotient_minpoly_map | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"adjoin_root.aeval_eq",
"adjoin_root.algebra_map_eq",
"adjoin_root.equiv'",
"adjoin_root.mk_self",
"adjoin_root.of",
"adjoin_root.quot_equiv_quot_map",
"alg_equiv.coe_ring_equiv",
"alg_equiv.coe_ring_hom_commutes",
"alg_equiv.commutes",
"alg_equiv.to_ring_equiv_eq_coe",
"alg_hom.comp_algebra_map... | Let `α` have minimal polynomial `f` over `R` and `I` be an ideal of `R`,
then `R[α] / (I) = (R[x] / (f)) / pS = (R/p)[x] / (f mod p)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_equiv_quotient_minpoly_map_apply_mk (pb : power_basis R S) (I : ideal R)
(g : R[X]) : pb.quotient_equiv_quotient_minpoly_map I
(ideal.quotient.mk _ (aeval pb.gen g)) = ideal.quotient.mk _ (g.map I^.quotient.mk) | by rw [power_basis.quotient_equiv_quotient_minpoly_map, alg_equiv.trans_apply,
alg_equiv.of_ring_equiv_apply, quotient_equiv_mk, alg_equiv.coe_ring_equiv',
adjoin_root.equiv'_symm_apply, power_basis.lift_aeval,
adjoin_root.aeval_eq, adjoin_root.quot_equiv_quot_map_apply_mk] | lemma | power_basis.quotient_equiv_quotient_minpoly_map_apply_mk | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"adjoin_root.aeval_eq",
"adjoin_root.quot_equiv_quot_map_apply_mk",
"alg_equiv.coe_ring_equiv'",
"alg_equiv.trans_apply",
"ideal",
"ideal.quotient.mk",
"power_basis",
"power_basis.lift_aeval",
"power_basis.quotient_equiv_quotient_minpoly_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_equiv_quotient_minpoly_map_symm_apply_mk (pb : power_basis R S) (I : ideal R)
(g : R[X]) : (pb.quotient_equiv_quotient_minpoly_map I).symm
(ideal.quotient.mk _ (g.map I^.quotient.mk)) = (ideal.quotient.mk _ (aeval pb.gen g)) | begin simp only [quotient_equiv_quotient_minpoly_map, to_ring_equiv_eq_coe, symm_trans_apply,
quot_equiv_quot_map_symm_apply_mk, of_ring_equiv_symm_apply, quotient_equiv_symm_mk,
to_ring_equiv_symm, ring_equiv.symm_symm, adjoin_root.equiv'_apply, coe_ring_equiv,
lift_hom_mk, symm_to_ring_equiv],
end | lemma | power_basis.quotient_equiv_quotient_minpoly_map_symm_apply_mk | ring_theory | src/ring_theory/adjoin_root.lean | [
"algebra.algebra.basic",
"data.polynomial.field_division",
"field_theory.minpoly.basic",
"ring_theory.adjoin.basic",
"ring_theory.finite_presentation",
"ring_theory.finite_type",
"ring_theory.power_basis",
"ring_theory.principal_ideal_domain",
"ring_theory.quotient_noetherian"
] | [
"ideal",
"ideal.quotient.mk",
"power_basis",
"ring_equiv.symm_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic (x : A) : Prop | ∃ p : R[X], p ≠ 0 ∧ aeval x p = 0 | def | is_algebraic | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [] | An element of an R-algebra is algebraic over R if it is a root of a nonzero polynomial
with coefficients in R. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
transcendental (x : A) : Prop | ¬ is_algebraic R x | def | transcendental | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic"
] | An element of an R-algebra is transcendental over R if it is not algebraic over R. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_transcendental_of_subsingleton [subsingleton R] (x : A) : transcendental R x | λ ⟨p, h, _⟩, h $ subsingleton.elim p 0 | lemma | is_transcendental_of_subsingleton | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"transcendental"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.is_algebraic (S : subalgebra R A) : Prop | ∀ x ∈ S, is_algebraic R x | def | subalgebra.is_algebraic | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic",
"subalgebra"
] | A subalgebra is algebraic if all its elements are algebraic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.is_algebraic : Prop | ∀ x : A, is_algebraic R x | def | algebra.is_algebraic | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic"
] | An algebra is algebraic if all its elements are algebraic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.is_algebraic_iff (S : subalgebra R A) :
S.is_algebraic ↔ @algebra.is_algebraic R S _ _ S.algebra | begin
delta algebra.is_algebraic subalgebra.is_algebraic,
rw subtype.forall',
refine forall_congr (λ x, exists_congr (λ p, and_congr iff.rfl _)),
have h : function.injective S.val := subtype.val_injective,
conv_rhs { rw [← h.eq_iff, alg_hom.map_zero] },
rw [← aeval_alg_hom_apply, S.val_apply]
end | lemma | subalgebra.is_algebraic_iff | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"alg_hom.map_zero",
"algebra.is_algebraic",
"subalgebra",
"subalgebra.is_algebraic",
"subtype.forall'",
"subtype.val_injective"
] | A subalgebra is algebraic if and only if it is algebraic as an algebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.is_algebraic_iff : algebra.is_algebraic R A ↔ (⊤ : subalgebra R A).is_algebraic | begin
delta algebra.is_algebraic subalgebra.is_algebraic,
simp only [algebra.mem_top, forall_prop_of_true, iff_self],
end | lemma | algebra.is_algebraic_iff | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra.is_algebraic",
"algebra.mem_top",
"forall_prop_of_true",
"is_algebraic",
"subalgebra",
"subalgebra.is_algebraic"
] | An algebra is algebraic if and only if it is algebraic as a subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_algebraic_iff_not_injective {x : A} : is_algebraic R x ↔
¬ function.injective (polynomial.aeval x : R[X] →ₐ[R] A) | by simp only [is_algebraic, injective_iff_map_eq_zero, not_forall, and.comm, exists_prop] | lemma | is_algebraic_iff_not_injective | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"exists_prop",
"is_algebraic",
"not_forall",
"polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral.is_algebraic [nontrivial R] {x : A} : is_integral R x → is_algebraic R x | λ ⟨p, hp, hpx⟩, ⟨p, hp.ne_zero, hpx⟩ | lemma | is_integral.is_algebraic | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic",
"is_integral",
"nontrivial"
] | An integral element of an algebra is algebraic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_algebraic_zero [nontrivial R] : is_algebraic R (0 : A) | ⟨_, X_ne_zero, aeval_X 0⟩ | lemma | is_algebraic_zero | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_algebra_map [nontrivial R] (x : R) : is_algebraic R (algebra_map R A x) | ⟨_, X_sub_C_ne_zero x, by rw [_root_.map_sub, aeval_X, aeval_C, sub_self]⟩ | lemma | is_algebraic_algebra_map | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map",
"is_algebraic",
"nontrivial"
] | An element of `R` is algebraic, when viewed as an element of the `R`-algebra `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_algebraic_one [nontrivial R] : is_algebraic R (1 : A) | by { rw ←_root_.map_one _, exact is_algebraic_algebra_map 1 } | lemma | is_algebraic_one | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic",
"is_algebraic_algebra_map",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_nat [nontrivial R] (n : ℕ) : is_algebraic R (n : A) | by { rw ←map_nat_cast _, exact is_algebraic_algebra_map n } | lemma | is_algebraic_nat | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic",
"is_algebraic_algebra_map",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_int [nontrivial R] (n : ℤ) : is_algebraic R (n : A) | by { rw ←_root_.map_int_cast (algebra_map R A), exact is_algebraic_algebra_map n } | lemma | is_algebraic_int | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map",
"is_algebraic",
"is_algebraic_algebra_map",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_rat (R : Type u) {A : Type v} [division_ring A] [field R] [algebra R A] (n : ℚ) :
is_algebraic R (n : A) | by { rw ←map_rat_cast (algebra_map R A), exact is_algebraic_algebra_map n } | lemma | is_algebraic_rat | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"algebra_map",
"division_ring",
"field",
"is_algebraic",
"is_algebraic_algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_of_mem_root_set {R : Type u} {A : Type v} [field R] [field A] [algebra R A]
{p : R[X]} {x : A} (hx : x ∈ p.root_set A) : is_algebraic R x | ⟨p, ne_zero_of_mem_root_set hx, aeval_eq_zero_of_mem_root_set hx⟩ | lemma | is_algebraic_of_mem_root_set | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"field",
"is_algebraic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_algebra_map_of_is_algebraic {a : S} :
is_algebraic R a → is_algebraic R (algebra_map S A a) | λ ⟨f, hf₁, hf₂⟩, ⟨f, hf₁, by rw [aeval_algebra_map_apply, hf₂, map_zero]⟩ | lemma | is_algebraic_algebra_map_of_is_algebraic | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map",
"is_algebraic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_alg_hom_of_is_algebraic {B} [ring B] [algebra R B]
(f : A →ₐ[R] B) {a : A} (h : is_algebraic R a) : is_algebraic R (f a) | let ⟨p, hp, ha⟩ := h in ⟨p, hp, by rw [aeval_alg_hom, f.comp_apply, ha, map_zero]⟩ | lemma | is_algebraic_alg_hom_of_is_algebraic | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"is_algebraic",
"ring"
] | This is slightly more general than `is_algebraic_algebra_map_of_is_algebraic` in that it
allows noncommutative intermediate rings `A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.alg_equiv.is_algebraic {B} [ring B] [algebra R B] (e : A ≃ₐ[R] B)
(h : algebra.is_algebraic R A) : algebra.is_algebraic R B | λ b, by convert ← is_algebraic_alg_hom_of_is_algebraic e.to_alg_hom (h _); apply e.apply_symm_apply | lemma | alg_equiv.is_algebraic | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"algebra.is_algebraic",
"is_algebraic_alg_hom_of_is_algebraic",
"ring"
] | Transfer `algebra.is_algebraic` across an `alg_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.alg_equiv.is_algebraic_iff {B} [ring B] [algebra R B] (e : A ≃ₐ[R] B) :
algebra.is_algebraic R A ↔ algebra.is_algebraic R B | ⟨e.is_algebraic, e.symm.is_algebraic⟩ | lemma | alg_equiv.is_algebraic_iff | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"algebra.is_algebraic",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_algebra_map_iff {a : S} (h : function.injective (algebra_map S A)) :
is_algebraic R (algebra_map S A a) ↔ is_algebraic R a | ⟨λ ⟨p, hp0, hp⟩, ⟨p, hp0, h (by rwa [map_zero, ← aeval_algebra_map_apply])⟩,
is_algebraic_algebra_map_of_is_algebraic⟩ | lemma | is_algebraic_algebra_map_iff | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map",
"is_algebraic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_of_pow {r : A} {n : ℕ} (hn : 0 < n) (ht : is_algebraic R (r ^ n)) :
is_algebraic R r | begin
obtain ⟨p, p_nonzero, hp⟩ := ht,
refine ⟨polynomial.expand _ n p, _, _⟩,
{ rwa polynomial.expand_ne_zero hn },
{ rwa polynomial.expand_aeval n p r },
end | lemma | is_algebraic_of_pow | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic",
"polynomial.expand_aeval",
"polynomial.expand_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
transcendental.pow {r : A} (ht : transcendental R r) {n : ℕ} (hn : 0 < n) :
transcendental R (r ^ n) | λ ht', ht $ is_algebraic_of_pow hn ht' | lemma | transcendental.pow | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic_of_pow",
"transcendental"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_iff_is_integral {x : A} :
is_algebraic K x ↔ is_integral K x | begin
refine ⟨_, is_integral.is_algebraic K⟩,
rintro ⟨p, hp, hpx⟩,
refine ⟨_, monic_mul_leading_coeff_inv hp, _⟩,
rw [← aeval_def, alg_hom.map_mul, hpx, zero_mul],
end | lemma | is_algebraic_iff_is_integral | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"alg_hom.map_mul",
"is_algebraic",
"is_integral",
"is_integral.is_algebraic",
"zero_mul"
] | An element of an algebra over a field is algebraic if and only if it is integral. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.is_algebraic_iff_is_integral :
algebra.is_algebraic K A ↔ algebra.is_integral K A | ⟨λ h x, is_algebraic_iff_is_integral.mp (h x),
λ h x, is_algebraic_iff_is_integral.mpr (h x)⟩ | lemma | algebra.is_algebraic_iff_is_integral | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra.is_algebraic",
"algebra.is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_trans (L_alg : is_algebraic K L) (A_alg : is_algebraic L A) :
is_algebraic K A | begin
simp only [is_algebraic, is_algebraic_iff_is_integral] at L_alg A_alg ⊢,
exact is_integral_trans L_alg A_alg,
end | lemma | algebra.is_algebraic_trans | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"is_algebraic",
"is_algebraic_iff_is_integral",
"is_integral_trans"
] | If L is an algebraic field extension of K and A is an algebraic algebra over L,
then A is algebraic over K. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_algebraic_of_larger_base_of_injective (hinj : function.injective (algebra_map R S))
{x : A} (A_alg : _root_.is_algebraic R x) : _root_.is_algebraic S x | let ⟨p, hp₁, hp₂⟩ := A_alg in
⟨p.map (algebra_map _ _),
by rwa [ne.def, ← degree_eq_bot, degree_map_eq_of_injective hinj, degree_eq_bot],
by simpa⟩ | lemma | is_algebraic_of_larger_base_of_injective | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map"
] | If x is algebraic over R, then x is algebraic over S when S is an extension of R,
and the map from `R` to `S` is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_algebraic_of_larger_base_of_injective (hinj : function.injective (algebra_map R S))
(A_alg : is_algebraic R A) : is_algebraic S A | λ x, is_algebraic_of_larger_base_of_injective hinj (A_alg x) | lemma | algebra.is_algebraic_of_larger_base_of_injective | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map",
"is_algebraic",
"is_algebraic_of_larger_base_of_injective"
] | If A is an algebraic algebra over R, then A is algebraic over S when S is an extension of R,
and the map from `R` to `S` is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_algebraic_of_larger_base {x : A} (A_alg : _root_.is_algebraic K x) :
_root_.is_algebraic L x | _root_.is_algebraic_of_larger_base_of_injective (algebra_map K L).injective A_alg | lemma | is_algebraic_of_larger_base | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map"
] | If x is a algebraic over K, then x is algebraic over L when L is an extension of K | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_algebraic_of_larger_base (A_alg : is_algebraic K A) : is_algebraic L A | is_algebraic_of_larger_base_of_injective (algebra_map K L).injective A_alg | lemma | algebra.is_algebraic_of_larger_base | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map",
"is_algebraic",
"is_algebraic_of_larger_base",
"is_algebraic_of_larger_base_of_injective"
] | If A is an algebraic algebra over K, then A is algebraic over L when L is an extension of K | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_of_finite [finite_dimensional K L] : algebra.is_integral K L | λ x, is_integral_of_submodule_noetherian ⊤
(is_noetherian.iff_fg.2 infer_instance) x algebra.mem_top | lemma | algebra.is_integral_of_finite | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra.is_integral",
"algebra.mem_top",
"finite_dimensional",
"is_integral_of_submodule_noetherian"
] | A field extension is integral if it is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_algebraic_of_finite [finite : finite_dimensional K L] : is_algebraic K L | algebra.is_algebraic_iff_is_integral.mpr (is_integral_of_finite K L) | lemma | algebra.is_algebraic_of_finite | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"finite",
"finite_dimensional",
"is_algebraic"
] | A field extension is algebraic if it is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_algebraic.alg_hom_bijective
(ha : algebra.is_algebraic K L) (f : L →ₐ[K] L) : function.bijective f | begin
refine ⟨f.to_ring_hom.injective, λ b, _⟩,
obtain ⟨p, hp, he⟩ := ha b,
let f' : p.root_set L → p.root_set L := (root_set_maps_to' id f).restrict f _ _,
have : function.surjective f' := finite.injective_iff_surjective.1
(λ _ _ h, subtype.eq $ f.to_ring_hom.injective $ subtype.ext_iff.1 h),
obtain ⟨a, ... | theorem | algebra.is_algebraic.alg_hom_bijective | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra.is_algebraic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.alg_hom.bijective [finite_dimensional K L] (ϕ : L →ₐ[K] L) : function.bijective ϕ | (algebra.is_algebraic_of_finite K L).alg_hom_bijective ϕ | theorem | alg_hom.bijective | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra.is_algebraic_of_finite",
"finite_dimensional"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic.alg_equiv_equiv_alg_hom
(ha : algebra.is_algebraic K L) : (L ≃ₐ[K] L) ≃* (L →ₐ[K] L) | { to_fun := λ ϕ, ϕ.to_alg_hom,
inv_fun := λ ϕ, alg_equiv.of_bijective ϕ (ha.alg_hom_bijective ϕ),
left_inv := λ _, by {ext, refl},
right_inv := λ _, by {ext, refl},
map_mul' := λ _ _, rfl } | def | algebra.is_algebraic.alg_equiv_equiv_alg_hom | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"alg_equiv.of_bijective",
"algebra.is_algebraic",
"inv_fun"
] | Bijection between algebra equivalences and algebra homomorphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.alg_equiv_equiv_alg_hom [finite_dimensional K L] :
(L ≃ₐ[K] L) ≃* (L →ₐ[K] L) | (algebra.is_algebraic_of_finite K L).alg_equiv_equiv_alg_hom K L | def | alg_equiv_equiv_alg_hom | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra.is_algebraic_of_finite",
"finite_dimensional"
] | Bijection between algebra equivalences and algebra homomorphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_integral_multiple [algebra R S] {z : S} (hz : is_algebraic R z)
(inj : ∀ x, algebra_map R S x = 0 → x = 0) :
∃ (x : integral_closure R S) (y ≠ (0 : R)),
z * algebra_map R S y = x | begin
rcases hz with ⟨p, p_ne_zero, px⟩,
set a := p.leading_coeff with a_def,
have a_ne_zero : a ≠ 0 := mt polynomial.leading_coeff_eq_zero.mp p_ne_zero,
have y_integral : is_integral R (algebra_map R S a) := is_integral_algebra_map,
have x_integral : is_integral R (z * algebra_map R S a) :=
⟨p.integral_n... | lemma | exists_integral_multiple | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"algebra_map",
"integral_closure",
"is_algebraic",
"is_integral",
"is_integral_algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_closure.exists_smul_eq_mul {L : Type*} [field L]
[algebra R S] [algebra S L] [algebra R L] [is_scalar_tower R S L] [is_integral_closure S R L]
(h : algebra.is_algebraic R L) (inj : function.injective (algebra_map R L))
(a : S) {b : S} (hb : b ≠ 0) : ∃ (c : S) (d ≠ (0 : R)), d • a = b * c | begin
obtain ⟨c, d, d_ne, hx⟩ := exists_integral_multiple
(h (algebra_map _ L a / algebra_map _ L b))
((injective_iff_map_eq_zero _).mp inj),
refine ⟨is_integral_closure.mk' S (c : L) c.2, d, d_ne,
is_integral_closure.algebra_map_injective S R L _⟩,
simp only [algebra.smul_def, ring_hom.map_mul, is_in... | lemma | is_integral_closure.exists_smul_eq_mul | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"algebra.is_algebraic",
"algebra.smul_def",
"algebra_map",
"exists_integral_multiple",
"field",
"is_integral_closure",
"is_integral_closure.algebra_map_mk'",
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply",
"mul_assoc",
"mul_comm",
"mul_div_cancel'",
"ring_hom.map_mul"
] | A fraction `(a : S) / (b : S)` can be reduced to `(c : S) / (d : R)`,
if `S` is the integral closure of `R` in an algebraic extension `L` of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_eq_of_aeval_div_X_ne_zero {x : L} {p : K[X]}
(aeval_ne : aeval x (div_X p) ≠ 0) :
x⁻¹ = aeval x (div_X p) / (aeval x p - algebra_map _ _ (p.coeff 0)) | begin
rw [inv_eq_iff_eq_inv, inv_div, eq_comm, div_eq_iff, sub_eq_iff_eq_add, mul_comm],
conv_lhs { rw ← div_X_mul_X_add p },
rw [alg_hom.map_add, alg_hom.map_mul, aeval_X, aeval_C],
exact aeval_ne
end | lemma | inv_eq_of_aeval_div_X_ne_zero | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"alg_hom.map_add",
"alg_hom.map_mul",
"algebra_map",
"div_eq_iff",
"inv_div",
"inv_eq_iff_eq_inv",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_root_of_coeff_zero_ne_zero {x : L} {p : K[X]}
(aeval_eq : aeval x p = 0) (coeff_zero_ne : p.coeff 0 ≠ 0) :
x⁻¹ = - (aeval x (div_X p) / algebra_map _ _ (p.coeff 0)) | begin
convert inv_eq_of_aeval_div_X_ne_zero (mt (λ h, (algebra_map K L).injective _) coeff_zero_ne),
{ rw [aeval_eq, zero_sub, div_neg] },
rw ring_hom.map_zero,
convert aeval_eq,
conv_rhs { rw ← div_X_mul_X_add p },
rw [alg_hom.map_add, alg_hom.map_mul, h, zero_mul, zero_add, aeval_C]
end | lemma | inv_eq_of_root_of_coeff_zero_ne_zero | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"alg_hom.map_add",
"alg_hom.map_mul",
"algebra_map",
"div_neg",
"inv_eq_of_aeval_div_X_ne_zero",
"ring_hom.map_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.inv_mem_of_root_of_coeff_zero_ne_zero {x : A} {p : K[X]}
(aeval_eq : aeval x p = 0) (coeff_zero_ne : p.coeff 0 ≠ 0) : (x⁻¹ : L) ∈ A | begin
suffices : (x⁻¹ : L) = (-p.coeff 0)⁻¹ • aeval x (div_X p),
{ rw [this], exact A.smul_mem (aeval x _).2 _ },
have : aeval (x : L) p = 0, by rw [subalgebra.aeval_coe, aeval_eq, subalgebra.coe_zero],
rw [inv_eq_of_root_of_coeff_zero_ne_zero this coeff_zero_ne, div_eq_inv_mul,
algebra.smul_def, map_inv₀, ... | lemma | subalgebra.inv_mem_of_root_of_coeff_zero_ne_zero | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra.smul_def",
"div_eq_inv_mul",
"inv_eq_of_root_of_coeff_zero_ne_zero",
"inv_neg",
"map_inv₀",
"neg_mul",
"subalgebra.aeval_coe",
"subalgebra.coe_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.inv_mem_of_algebraic {x : A} (hx : is_algebraic K (x : L)) : (x⁻¹ : L) ∈ A | begin
obtain ⟨p, ne_zero, aeval_eq⟩ := hx,
rw [subalgebra.aeval_coe, subalgebra.coe_eq_zero] at aeval_eq,
revert ne_zero aeval_eq,
refine p.rec_on_horner _ _ _,
{ intro h,
contradiction },
{ intros p a hp ha ih ne_zero aeval_eq,
refine A.inv_mem_of_root_of_coeff_zero_ne_zero aeval_eq _,
rwa [coe... | lemma | subalgebra.inv_mem_of_algebraic | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"alg_hom.map_mul",
"ih",
"inv_zero",
"is_algebraic",
"mul_eq_zero",
"ne_zero",
"subalgebra.aeval_coe",
"subalgebra.coe_eq_zero",
"subalgebra.coe_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.is_field_of_algebraic (hKL : algebra.is_algebraic K L) : is_field A | { mul_inv_cancel := λ a ha, ⟨
⟨a⁻¹, A.inv_mem_of_algebraic (hKL a)⟩,
subtype.ext (mul_inv_cancel (mt (subalgebra.coe_eq_zero _).mp ha))⟩,
.. show nontrivial A, by apply_instance,
.. subalgebra.to_comm_ring A } | lemma | subalgebra.is_field_of_algebraic | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra.is_algebraic",
"is_field",
"mul_inv_cancel",
"nontrivial",
"subalgebra.coe_eq_zero",
"subalgebra.to_comm_ring",
"subtype.ext"
] | In an algebraic extension L/K, an intermediate subalgebra is a field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial.has_smul_pi [semiring R'] [has_smul R' S'] :
has_smul (R'[X]) (R' → S') | ⟨λ p f x, eval x p • f x⟩ | def | polynomial.has_smul_pi | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"has_smul",
"semiring"
] | This is not an instance as it forms a diamond with `pi.has_smul`.
See the `instance_diamonds` test for details. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial.has_smul_pi' [comm_semiring R'] [semiring S'] [algebra R' S']
[has_smul S' T'] :
has_smul (R'[X]) (S' → T') | ⟨λ p f x, aeval x p • f x⟩ | def | polynomial.has_smul_pi' | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"comm_semiring",
"has_smul",
"semiring"
] | This is not an instance as it forms a diamond with `pi.has_smul`.
See the `instance_diamonds` test for details. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial_smul_apply [semiring R'] [has_smul R' S']
(p : R'[X]) (f : R' → S') (x : R') :
(p • f) x = eval x p • f x | rfl | lemma | polynomial_smul_apply | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"has_smul",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_smul_apply' [comm_semiring R'] [semiring S'] [algebra R' S']
[has_smul S' T'] (p : R'[X]) (f : S' → T') (x : S') :
(p • f) x = aeval x p • f x | rfl | lemma | polynomial_smul_apply' | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"comm_semiring",
"has_smul",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial.algebra_pi :
algebra (R'[X]) (S' → T') | { to_fun := λ p z, algebra_map S' T' (aeval z p),
map_one' := funext $ λ z, by simp only [polynomial.aeval_one, pi.one_apply, map_one],
map_mul' := λ f g, funext $ λ z, by simp only [pi.mul_apply, map_mul],
map_zero' := funext $ λ z, by simp only [polynomial.aeval_zero, pi.zero_apply, map_zero],
map_add' := λ f... | def | polynomial.algebra_pi | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra",
"algebra.algebra_map_eq_smul_one",
"algebra.smul_mul_assoc",
"algebra_map",
"map_mul",
"map_one",
"mul_comm",
"one_mul",
"pi.mul_apply",
"pi.one_apply",
"polynomial.aeval_add",
"polynomial.aeval_one",
"polynomial.aeval_zero",
"polynomial.has_smul_pi'",
"polynomial_smul_apply'"... | This is not an instance for the same reasons as `polynomial.has_smul_pi'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial.algebra_map_pi_eq_aeval :
(algebra_map (R'[X]) (S' → T') : R'[X] → (S' → T')) =
λ p z, algebra_map _ _ (aeval z p) | rfl | lemma | polynomial.algebra_map_pi_eq_aeval | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial.algebra_map_pi_self_eq_eval :
(algebra_map (R'[X]) (R' → R') : R'[X] → (R' → R')) = λ p z, eval z p | rfl | lemma | polynomial.algebra_map_pi_self_eq_eval | ring_theory | src/ring_theory/algebraic.lean | [
"linear_algebra.finite_dimensional",
"ring_theory.integral_closure",
"data.polynomial.integral_normalization"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent : Prop | injective (mv_polynomial.aeval x : mv_polynomial ι R →ₐ[R] A) | def | algebraic_independent | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"mv_polynomial",
"mv_polynomial.aeval"
] | `algebraic_independent R x` states the family of elements `x`
is algebraically independent over `R`, meaning that the canonical
map out of the multivariable polynomial ring is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebraic_independent_iff_ker_eq_bot : algebraic_independent R x ↔
(mv_polynomial.aeval x : mv_polynomial ι R →ₐ[R] A).to_ring_hom.ker = ⊥ | ring_hom.injective_iff_ker_eq_bot _ | theorem | algebraic_independent_iff_ker_eq_bot | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"mv_polynomial",
"mv_polynomial.aeval",
"ring_hom.injective_iff_ker_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_iff : algebraic_independent R x ↔
∀p : mv_polynomial ι R, mv_polynomial.aeval (x : ι → A) p = 0 → p = 0 | injective_iff_map_eq_zero _ | theorem | algebraic_independent_iff | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"mv_polynomial",
"mv_polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.eq_zero_of_aeval_eq_zero (h : algebraic_independent R x) :
∀p : mv_polynomial ι R, mv_polynomial.aeval (x : ι → A) p = 0 → p = 0 | algebraic_independent_iff.1 h | theorem | algebraic_independent.eq_zero_of_aeval_eq_zero | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"mv_polynomial",
"mv_polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_iff_injective_aeval :
algebraic_independent R x ↔ injective (mv_polynomial.aeval x : mv_polynomial ι R →ₐ[R] A) | iff.rfl | theorem | algebraic_independent_iff_injective_aeval | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"mv_polynomial",
"mv_polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_empty_type_iff [is_empty ι] :
algebraic_independent R x ↔ injective (algebra_map R A) | have aeval x = (algebra.of_id R A).comp (@is_empty_alg_equiv R ι _ _).to_alg_hom,
by { ext i, exact is_empty.elim' ‹is_empty ι› i },
begin
rw [algebraic_independent, this,
← injective.of_comp_iff' _ (@is_empty_alg_equiv R ι _ _).bijective],
refl
end | lemma | algebraic_independent_empty_type_iff | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebra.of_id",
"algebra_map",
"algebraic_independent",
"is_empty",
"is_empty.elim'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_injective : injective (algebra_map R A) | by simpa [← mv_polynomial.algebra_map_eq, function.comp] using
(injective.of_comp_iff
(algebraic_independent_iff_injective_aeval.1 hx) (mv_polynomial.C)).2
(mv_polynomial.C_injective _ _) | lemma | algebraic_independent.algebra_map_injective | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebra_map",
"algebra_map_injective",
"mv_polynomial.C",
"mv_polynomial.C_injective",
"mv_polynomial.algebra_map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent : linear_independent R x | begin
rw [linear_independent_iff_injective_total],
have : finsupp.total ι A R x =
(mv_polynomial.aeval x).to_linear_map.comp (finsupp.total ι _ R X),
{ ext, simp },
rw this,
refine hx.comp _,
rw [← linear_independent_iff_injective_total],
exact linear_independent_X _ _
end | lemma | algebraic_independent.linear_independent | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"finsupp.total",
"linear_independent",
"linear_independent_iff_injective_total",
"mv_polynomial.aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective [nontrivial R] : injective x | hx.linear_independent.injective | lemma | algebraic_independent.injective | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero [nontrivial R] (i : ι) : x i ≠ 0 | hx.linear_independent.ne_zero i | lemma | algebraic_independent.ne_zero | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"ne_zero",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : ι' → ι) (hf : function.injective f) : algebraic_independent R (x ∘ f) | λ p q, by simpa [aeval_rename, (rename_injective f hf).eq_iff] using @hx (rename f p) (rename f q) | lemma | algebraic_independent.comp | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_range : algebraic_independent R (coe : range x → A) | by simpa using hx.comp _ (range_splitting_injective x) | lemma | algebraic_independent.coe_range | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map {f : A →ₐ[R] A'} (hf_inj : set.inj_on f (adjoin R (range x))) :
algebraic_independent R (f ∘ x) | have aeval (f ∘ x) = f.comp (aeval x), by ext; simp,
have h : ∀ p : mv_polynomial ι R, aeval x p ∈ (@aeval R _ _ _ _ _ (coe : range x → A)).range,
{ intro p,
rw [alg_hom.mem_range],
refine ⟨mv_polynomial.rename (cod_restrict x (range x) (mem_range_self)) p, _⟩,
simp [function.comp, aeval_rename] },
begin
... | lemma | algebraic_independent.map | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"alg_hom.mem_range",
"algebraic_independent",
"mv_polynomial",
"set.inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map' {f : A →ₐ[R] A'} (hf_inj : injective f) : algebraic_independent R (f ∘ x) | hx.map (inj_on_of_injective hf_inj _) | lemma | algebraic_independent.map' | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_comp (f : A →ₐ[R] A') (hfv : algebraic_independent R (f ∘ x)) :
algebraic_independent R x | have aeval (f ∘ x) = f.comp (aeval x), by ext; simp,
by rw [algebraic_independent, this] at hfv; exact hfv.of_comp | lemma | algebraic_independent.of_comp | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom.algebraic_independent_iff (f : A →ₐ[R] A') (hf : injective f) :
algebraic_independent R (f ∘ x) ↔ algebraic_independent R x | ⟨λ h, h.of_comp f, λ h, h.map (inj_on_of_injective hf _)⟩ | lemma | alg_hom.algebraic_independent_iff | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_of_subsingleton [subsingleton R] : algebraic_independent R x | algebraic_independent_iff.2 (λ l hl, subsingleton.elim _ _) | lemma | algebraic_independent_of_subsingleton | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_equiv (e : ι ≃ ι') {f : ι' → A} :
algebraic_independent R (f ∘ e) ↔ algebraic_independent R f | ⟨λ h, function.comp.right_id f ▸ e.self_comp_symm ▸ h.comp _ e.symm.injective,
λ h, h.comp _ e.injective⟩ | theorem | algebraic_independent_equiv | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_equiv' (e : ι ≃ ι') {f : ι' → A} {g : ι → A} (h : f ∘ e = g) :
algebraic_independent R g ↔ algebraic_independent R f | h ▸ algebraic_independent_equiv e | theorem | algebraic_independent_equiv' | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_subtype_range {ι} {f : ι → A} (hf : injective f) :
algebraic_independent R (coe : range f → A) ↔ algebraic_independent R f | iff.symm $ algebraic_independent_equiv' (equiv.of_injective f hf) rfl | theorem | algebraic_independent_subtype_range | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_equiv'",
"equiv.of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_image {ι} {s : set ι} {f : ι → A} (hf : set.inj_on f s) :
algebraic_independent R (λ x : s, f x) ↔ algebraic_independent R (λ x : f '' s, (x : A)) | algebraic_independent_equiv' (equiv.set.image_of_inj_on _ _ hf) rfl | theorem | algebraic_independent_image | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_equiv'",
"equiv.set.image_of_inj_on",
"set.inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_adjoin (hs : algebraic_independent R x) :
@algebraic_independent ι R (adjoin R (range x))
(λ i : ι, ⟨x i, subset_adjoin (mem_range_self i)⟩) _ _ _ | algebraic_independent.of_comp (adjoin R (range x)).val hs | lemma | algebraic_independent_adjoin | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent.of_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.restrict_scalars {K : Type*} [comm_ring K] [algebra R K]
[algebra K A] [is_scalar_tower R K A]
(hinj : function.injective (algebra_map R K)) (ai : algebraic_independent K x) :
algebraic_independent R x | have (aeval x : mv_polynomial ι K →ₐ[K] A).to_ring_hom.comp
(mv_polynomial.map (algebra_map R K)) =
(aeval x : mv_polynomial ι R →ₐ[R] A).to_ring_hom,
by { ext; simp [algebra_map_eq_smul_one] },
begin
show injective (aeval x).to_ring_hom,
rw [← this],
exact injective.comp ai (mv_polynomial.map_injective... | lemma | algebraic_independent.restrict_scalars | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebra",
"algebra_map",
"algebraic_independent",
"comm_ring",
"is_scalar_tower",
"mv_polynomial",
"mv_polynomial.map",
"mv_polynomial.map_injective"
] | A set of algebraically independent elements in an algebra `A` over a ring `K` is also
algebraically independent over a subring `R` of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebraic_independent_finset_map_embedding_subtype
(s : set A) (li : algebraic_independent R (coe : s → A)) (t : finset s) :
algebraic_independent R (coe : (finset.map (embedding.subtype s) t) → A) | begin
let f : t.map (embedding.subtype s) → s := λ x, ⟨x.1, begin
obtain ⟨x, h⟩ := x,
rw [finset.mem_map] at h,
obtain ⟨a, ha, rfl⟩ := h,
simp only [subtype.coe_prop, embedding.coe_subtype],
end⟩,
convert algebraic_independent.comp li f _,
rintros ⟨x, hx⟩ ⟨y, hy⟩,
rw [finset.mem_map] at hx hy,... | lemma | algebraic_independent_finset_map_embedding_subtype | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent.comp",
"finset",
"finset.map",
"finset.mem_map",
"imp_self",
"subtype.coe_prop",
"subtype.mk_eq_mk"
] | Every finite subset of an algebraically independent set is algebraically independent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebraic_independent_bounded_of_finset_algebraic_independent_bounded {n : ℕ}
(H : ∀ s : finset A, algebraic_independent R (λ i : s, (i : A)) → s.card ≤ n) :
∀ s : set A, algebraic_independent R (coe : s → A) → cardinal.mk s ≤ n | begin
intros s li,
apply cardinal.card_le_of,
intro t,
rw ← finset.card_map (embedding.subtype s),
apply H,
apply algebraic_independent_finset_map_embedding_subtype _ li,
end | lemma | algebraic_independent_bounded_of_finset_algebraic_independent_bounded | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_finset_map_embedding_subtype",
"cardinal.card_le_of",
"cardinal.mk",
"finset",
"finset.card_map"
] | If every finite set of algebraically independent element has cardinality at most `n`,
then the same is true for arbitrary sets of algebraically independent elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebraic_independent.restrict_of_comp_subtype {s : set ι}
(hs : algebraic_independent R (x ∘ coe : s → A)) :
algebraic_independent R (s.restrict x) | hs | lemma | algebraic_independent.restrict_of_comp_subtype | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_empty_iff : algebraic_independent R (λ x, x : (∅ : set A) → A) ↔
injective (algebra_map R A) | by simp | lemma | algebraic_independent_empty_iff | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebra_map",
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.mono {t s : set A} (h : t ⊆ s)
(hx : algebraic_independent R (λ x, x : s → A)) : algebraic_independent R (λ x, x : t → A) | by simpa [function.comp] using hx.comp (inclusion h) (inclusion_injective h) | lemma | algebraic_independent.mono | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.to_subtype_range {ι} {f : ι → A} (hf : algebraic_independent R f) :
algebraic_independent R (coe : range f → A) | begin
nontriviality R,
{ rwa algebraic_independent_subtype_range hf.injective }
end | theorem | algebraic_independent.to_subtype_range | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_subtype_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.to_subtype_range' {ι} {f : ι → A} (hf : algebraic_independent R f)
{t} (ht : range f = t) :
algebraic_independent R (coe : t → A) | ht ▸ hf.to_subtype_range | theorem | algebraic_independent.to_subtype_range' | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_comp_subtype {s : set ι} :
algebraic_independent R (x ∘ coe : s → A) ↔
∀ p ∈ (mv_polynomial.supported R s), aeval x p = 0 → p = 0 | have (aeval (x ∘ coe : s → A) : _ →ₐ[R] _) =
(aeval x).comp (rename coe), by ext; simp,
have ∀ p : mv_polynomial s R, rename (coe : s → ι) p = 0 ↔ p = 0,
from (injective_iff_map_eq_zero' (rename (coe : s → ι) : mv_polynomial s R →ₐ[R] _).to_ring_hom).1
(rename_injective _ subtype.val_injective),
by simp [algebr... | theorem | algebraic_independent_comp_subtype | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_iff",
"mv_polynomial",
"mv_polynomial.supported",
"subtype.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_subtype {s : set A} :
algebraic_independent R (λ x, x : s → A) ↔
∀ (p : mv_polynomial A R), p ∈ mv_polynomial.supported R s → aeval id p = 0 → p = 0 | by apply @algebraic_independent_comp_subtype _ _ _ id | theorem | algebraic_independent_subtype | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_comp_subtype",
"mv_polynomial",
"mv_polynomial.supported"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_of_finite (s : set A)
(H : ∀ t ⊆ s, t.finite → algebraic_independent R (λ x, x : t → A)) :
algebraic_independent R (λ x, x : s → A) | algebraic_independent_subtype.2 $
λ p hp, algebraic_independent_subtype.1 (H _ (mem_supported.1 hp) (finset.finite_to_set _)) _
(by simp) | lemma | algebraic_independent_of_finite | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"finset.finite_to_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.image_of_comp {ι ι'} (s : set ι) (f : ι → ι') (g : ι' → A)
(hs : algebraic_independent R (λ x : s, g (f x))) :
algebraic_independent R (λ x : f '' s, g x) | begin
nontriviality R,
have : inj_on f s, from inj_on_iff_injective.2 hs.injective.of_comp,
exact (algebraic_independent_equiv' (equiv.set.image_of_inj_on f s this) rfl).1 hs
end | theorem | algebraic_independent.image_of_comp | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_equiv'",
"equiv.set.image_of_inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent.image {ι} {s : set ι} {f : ι → A}
(hs : algebraic_independent R (λ x : s, f x)) : algebraic_independent R (λ x : f '' s, (x : A)) | by convert algebraic_independent.image_of_comp s f id hs | theorem | algebraic_independent.image | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent.image_of_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_Union_of_directed {η : Type*} [nonempty η]
{s : η → set A} (hs : directed (⊆) s)
(h : ∀ i, algebraic_independent R (λ x, x : s i → A)) :
algebraic_independent R (λ x, x : (⋃ i, s i) → A) | begin
refine algebraic_independent_of_finite (⋃ i, s i) (λ t ht ft, _),
rcases finite_subset_Union ft ht with ⟨I, fi, hI⟩,
rcases hs.finset_le fi.to_finset with ⟨i, hi⟩,
exact (h i).mono (subset.trans hI $ Union₂_subset $
λ j hj, hi j (fi.mem_to_finset.2 hj))
end | lemma | algebraic_independent_Union_of_directed | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_of_finite",
"directed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebraic_independent_sUnion_of_directed {s : set (set A)}
(hsn : s.nonempty)
(hs : directed_on (⊆) s)
(h : ∀ a ∈ s, algebraic_independent R (λ x, x : (a : set A) → A)) :
algebraic_independent R (λ x, x : (⋃₀ s) → A) | by letI : nonempty s := nonempty.to_subtype hsn;
rw sUnion_eq_Union; exact
algebraic_independent_Union_of_directed hs.directed_coe (by simpa using h) | lemma | algebraic_independent_sUnion_of_directed | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"algebraic_independent_Union_of_directed",
"directed_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_maximal_algebraic_independent
(s t : set A) (hst : s ⊆ t)
(hs : algebraic_independent R (coe : s → A)) :
∃ u : set A, algebraic_independent R (coe : u → A) ∧ s ⊆ u ∧ u ⊆ t ∧
∀ x : set A, algebraic_independent R (coe : x → A) →
u ⊆ x → x ⊆ t → x = u | begin
rcases zorn_subset_nonempty
{ u : set A | algebraic_independent R (coe : u → A) ∧ s ⊆ u ∧ u ⊆ t }
(λ c hc chainc hcn, ⟨⋃₀ c, begin
refine ⟨⟨algebraic_independent_sUnion_of_directed hcn
chainc.directed_on
(λ a ha, (hc ha).1), _, _⟩, _⟩,
{ cases hcn with x hx,
exact s... | lemma | exists_maximal_algebraic_independent | ring_theory | src/ring_theory/algebraic_independent.lean | [
"ring_theory.adjoin.basic",
"linear_algebra.linear_independent",
"ring_theory.mv_polynomial.basic",
"data.mv_polynomial.supported",
"ring_theory.algebraic",
"data.mv_polynomial.equiv"
] | [
"algebraic_independent",
"set.subset.refl",
"zorn_subset_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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