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 |
|---|---|---|---|---|---|---|---|---|---|---|
normalize_scale_roots_monic (h : p ≠ 0) : (normalize_scale_roots p).monic | begin
delta monic leading_coeff,
rw nat_degree_eq_of_degree_eq (normalize_scale_roots_degree p),
suffices : p = 0 → (0 : R) = 1,
{ simpa [normalize_scale_roots, coeff_monomial] },
exact λ h', (h h').rec _,
end | lemma | normalize_scale_roots_monic | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"normalize_scale_roots",
"normalize_scale_roots_degree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_elem_leading_coeff_mul (h : p.eval₂ f x = 0) :
f.is_integral_elem (f p.leading_coeff * x) | begin
by_cases h' : 1 ≤ p.nat_degree,
{ use normalize_scale_roots p,
have : p ≠ 0 := λ h'', by { rw [h'', nat_degree_zero] at h', exact nat.not_succ_le_zero 0 h' },
use normalize_scale_roots_monic p this,
rw [normalize_scale_roots_eval₂_leading_coeff_mul p h' f x, h, mul_zero] },
{ by_cases hp : p.map... | lemma | ring_hom.is_integral_elem_leading_coeff_mul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"mul_zero",
"nat.one_le_iff_ne_zero",
"normalize_scale_roots",
"normalize_scale_roots_eval₂_leading_coeff_mul",
"normalize_scale_roots_monic",
"not_not",
"zero_mul"
] | Given a `p : R[X]` and a `x : S` such that `p.eval₂ f x = 0`,
`f p.leading_coeff * x` is integral. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_leading_coeff_smul [algebra R S] (h : aeval x p = 0) :
is_integral R (p.leading_coeff • x) | begin
rw aeval_def at h,
rw algebra.smul_def,
exact (algebra_map R S).is_integral_elem_leading_coeff_mul p x h,
end | lemma | is_integral_leading_coeff_smul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"is_integral"
] | Given a `p : R[X]` and a root `x : S`,
then `p.leading_coeff • x : S` is integral over `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_closure (A R B : Type*) [comm_ring R] [comm_semiring A] [comm_ring B]
[algebra R B] [algebra A B] : Prop | (algebra_map_injective [] : function.injective (algebra_map A B))
(is_integral_iff : ∀ {x : B}, is_integral R x ↔ ∃ y, algebra_map A B y = x) | class | is_integral_closure | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra_map",
"algebra_map_injective",
"comm_ring",
"comm_semiring",
"is_integral"
] | `is_integral_closure 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_closure.is_integral_closure (R A : Type*) [comm_ring R] [comm_ring A]
[algebra R A] : is_integral_closure (integral_closure R A) R A | ⟨subtype.coe_injective, λ x, ⟨λ h, ⟨⟨x, h⟩, rfl⟩, by { rintro ⟨⟨_, h⟩, rfl⟩, exact h }⟩⟩ | instance | integral_closure.is_integral_closure | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"comm_ring",
"integral_closure",
"is_integral_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral [algebra R A] [is_scalar_tower R A B] (x : A) : is_integral R x | (is_integral_algebra_map_iff (algebra_map_injective A R B)).mp $
show is_integral R (algebra_map A B x), from is_integral_iff.mpr ⟨x, rfl⟩ | theorem | is_integral_closure.is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra_map",
"algebra_map_injective",
"is_integral",
"is_integral_algebra_map_iff",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_algebra [algebra R A] [is_scalar_tower R A B] :
algebra.is_integral R A | λ x, is_integral_closure.is_integral R B x | theorem | is_integral_closure.is_integral_algebra | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra.is_integral",
"is_integral_closure.is_integral",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_smul_divisors [algebra R A] [is_scalar_tower R A B] [no_zero_smul_divisors R B] :
no_zero_smul_divisors R A | begin
refine function.injective.no_zero_smul_divisors
_ (is_integral_closure.algebra_map_injective A R B) (map_zero _) (λ _ _, _),
simp only [algebra.algebra_map_eq_smul_one, is_scalar_tower.smul_assoc],
end | theorem | is_integral_closure.no_zero_smul_divisors | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra.algebra_map_eq_smul_one",
"function.injective.no_zero_smul_divisors",
"is_scalar_tower",
"no_zero_smul_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (x : B) (hx : is_integral R x) : A | classical.some (is_integral_iff.mp hx) | def | is_integral_closure.mk' | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"mk'"
] | If `x : B` is integral over `R`, then it is an element of the integral closure of `R` in `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_mk' (x : B) (hx : is_integral R x) :
algebra_map A B (mk' A x hx) = x | classical.some_spec (is_integral_iff.mp hx) | lemma | is_integral_closure.algebra_map_mk' | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_one (h : is_integral R (1 : B) := is_integral_one) :
mk' A 1 h = 1 | algebra_map_injective A R B $ by rw [algebra_map_mk', ring_hom.map_one] | lemma | is_integral_closure.mk'_one | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map_injective",
"is_integral",
"is_integral_one",
"mk'",
"mk'_one",
"ring_hom.map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_zero (h : is_integral R (0 : B) := is_integral_zero) :
mk' A 0 h = 0 | algebra_map_injective A R B $ by rw [algebra_map_mk', ring_hom.map_zero] | lemma | is_integral_closure.mk'_zero | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map_injective",
"is_integral",
"is_integral_zero",
"mk'",
"mk'_zero",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_add (x y : B) (hx : is_integral R x) (hy : is_integral R y) :
mk' A (x + y) (is_integral_add hx hy) = mk' A x hx + mk' A y hy | algebra_map_injective A R B $ by simp only [algebra_map_mk', ring_hom.map_add] | lemma | is_integral_closure.mk'_add | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map_injective",
"is_integral",
"is_integral_add",
"mk'",
"mk'_add",
"ring_hom.map_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mul (x y : B) (hx : is_integral R x) (hy : is_integral R y) :
mk' A (x * y) (is_integral_mul hx hy) = mk' A x hx * mk' A y hy | algebra_map_injective A R B $ by simp only [algebra_map_mk', ring_hom.map_mul] | lemma | is_integral_closure.mk'_mul | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map_injective",
"is_integral",
"is_integral_mul",
"mk'",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_algebra_map [algebra R A] [is_scalar_tower R A B] (x : R)
(h : is_integral R (algebra_map R B x) := is_integral_algebra_map) :
is_integral_closure.mk' A (algebra_map R B x) h = algebra_map R A x | algebra_map_injective A R B $ by rw [algebra_map_mk', ← is_scalar_tower.algebra_map_apply] | lemma | is_integral_closure.mk'_algebra_map | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra_map",
"algebra_map_injective",
"is_integral",
"is_integral_algebra_map",
"is_integral_closure.mk'",
"is_scalar_tower",
"is_scalar_tower.algebra_map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift : S →ₐ[R] A | { to_fun := λ x, mk' A (algebra_map S B x) (is_integral.algebra_map (h x)),
map_one' := by simp only [ring_hom.map_one, mk'_one],
map_zero' := by simp only [ring_hom.map_zero, mk'_zero],
map_add' := λ x y, by simp_rw [← mk'_add, ring_hom.map_add],
map_mul' := λ x y, by simp_rw [← mk'_mul, ring_hom.map_mul],
c... | def | is_integral_closure.lift | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral.algebra_map",
"is_scalar_tower.algebra_map_apply",
"lift",
"mk'",
"mk'_add",
"mk'_one",
"mk'_zero",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_one",
"ring_hom.map_zero"
] | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_lift (x : S) : algebra_map A B (lift A B h x) = algebra_map S B x | algebra_map_mk' _ _ _ | lemma | is_integral_closure.algebra_map_lift | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv : A ≃ₐ[R] A' | alg_equiv.of_alg_hom (lift _ B (is_integral_algebra R B)) (lift _ B (is_integral_algebra R B))
(by { ext x, apply algebra_map_injective A' R B, simp })
(by { ext x, apply algebra_map_injective A R B, simp }) | def | is_integral_closure.equiv | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"alg_equiv.of_alg_hom",
"algebra_map_injective",
"equiv",
"lift"
] | Integral closures are all isomorphic to each other. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_equiv (x : A) : algebra_map A' B (equiv R A B A' x) = algebra_map A B x | algebra_map_lift _ _ _ _ | lemma | is_integral_closure.algebra_map_equiv | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_trans_aux (x : B) {p : A[X]} (pmonic : monic p) (hp : aeval x p = 0) :
is_integral (adjoin R (↑(p.map $ algebra_map A B).frange : set B)) x | begin
generalize hS : (↑(p.map $ algebra_map A B).frange : set B) = S,
have coeffs_mem : ∀ i, (p.map $ algebra_map A B).coeff i ∈ adjoin R S,
{ intro i, by_cases hi : (p.map $ algebra_map A B).coeff i = 0,
{ rw hi, exact subalgebra.zero_mem _ },
rw ← hS,
exact subset_adjoin (coeff_mem_frange _ _ hi) }... | lemma | is_integral_trans_aux | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral",
"polynomial.mem_map_range",
"set.mem_range",
"subalgebra.zero_mem",
"subtype.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_trans (A_int : algebra.is_integral R A) (x : B) (hx : is_integral A x) :
is_integral R x | begin
rcases hx with ⟨p, pmonic, hp⟩,
let S : set B := ↑(p.map $ algebra_map A B).frange,
refine is_integral_of_mem_of_fg (adjoin R (S ∪ {x})) _ _ (subset_adjoin $ or.inr rfl),
refine fg_trans (fg_adjoin_of_finite (finset.finite_to_set _) (λ x hx, _)) _,
{ rw [finset.mem_coe, frange, finset.mem_image] at hx,
... | lemma | is_integral_trans | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.is_integral",
"algebra_map",
"fg_adjoin_of_finite",
"fg_adjoin_singleton_of_integral",
"finset.finite_to_set",
"finset.mem_coe",
"finset.mem_image",
"is_integral",
"is_integral_of_mem_of_fg",
"is_integral_trans_aux",
"is_scalar_tower.to_alg_hom",
"map_is_integral"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.is_integral_trans (hA : algebra.is_integral R A) (hB : algebra.is_integral A B) :
algebra.is_integral R B | λ x, is_integral_trans hA x (hB x) | lemma | algebra.is_integral_trans | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.is_integral",
"is_integral_trans"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.is_integral_trans (hf : f.is_integral) (hg : g.is_integral) :
(g.comp f).is_integral | @algebra.is_integral_trans R S T _ _ _ g.to_algebra (g.comp f).to_algebra f.to_algebra
(@is_scalar_tower.of_algebra_map_eq R S T _ _ _ f.to_algebra g.to_algebra (g.comp f).to_algebra
(ring_hom.comp_apply g f)) hf hg | lemma | ring_hom.is_integral_trans | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.is_integral_trans",
"is_integral",
"is_scalar_tower.of_algebra_map_eq",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_of_surjective (hf : function.surjective f) : f.is_integral | λ x, (hf x).rec_on (λ y hy, (hy ▸ f.is_integral_map : f.is_integral_elem x)) | lemma | ring_hom.is_integral_of_surjective | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_of_surjective (h : function.surjective (algebra_map R A)) :
algebra.is_integral R A | (algebra_map R A).is_integral_of_surjective h | lemma | is_integral_of_surjective | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.is_integral",
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_tower_bot_of_is_integral (H : function.injective (algebra_map A B))
{x : A} (h : is_integral R (algebra_map A B x)) : is_integral R x | begin
rcases h with ⟨p, ⟨hp, hp'⟩⟩,
refine ⟨p, ⟨hp, _⟩⟩,
rw [is_scalar_tower.algebra_map_eq R A B, ← eval₂_map,
eval₂_hom, ← ring_hom.map_zero (algebra_map A B)] at hp',
rw [eval₂_eq_eval_map],
exact H hp',
end | lemma | is_integral_tower_bot_of_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral",
"is_scalar_tower.algebra_map_eq",
"ring_hom.map_zero"
] | 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` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.is_integral_tower_bot_of_is_integral (hg : function.injective g)
(hfg : (g.comp f).is_integral) : f.is_integral | λ x,
@is_integral_tower_bot_of_is_integral R S T _ _ _ g.to_algebra (g.comp f).to_algebra f.to_algebra
(@is_scalar_tower.of_algebra_map_eq R S T _ _ _ f.to_algebra g.to_algebra (g.comp f).to_algebra
(ring_hom.comp_apply g f)) hg x (hfg (g x)) | lemma | ring_hom.is_integral_tower_bot_of_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"is_integral_tower_bot_of_is_integral",
"is_scalar_tower.of_algebra_map_eq",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_tower_bot_of_is_integral_field {R A B : Type*} [comm_ring R] [field A]
[comm_ring B] [nontrivial B] [algebra R A] [algebra A B] [algebra R B] [is_scalar_tower R A B]
{x : A} (h : is_integral R (algebra_map A B x)) : is_integral R x | is_integral_tower_bot_of_is_integral (algebra_map A B).injective h | lemma | is_integral_tower_bot_of_is_integral_field | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra_map",
"comm_ring",
"field",
"is_integral",
"is_integral_tower_bot_of_is_integral",
"is_scalar_tower",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_elem_of_is_integral_elem_comp {x : T}
(h : (g.comp f).is_integral_elem x) : g.is_integral_elem x | let ⟨p, ⟨hp, hp'⟩⟩ := h in ⟨p.map f, hp.map f, by rwa ← eval₂_map at hp'⟩ | lemma | ring_hom.is_integral_elem_of_is_integral_elem_comp | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.is_integral_tower_top_of_is_integral (h : (g.comp f).is_integral) : g.is_integral | λ x, ring_hom.is_integral_elem_of_is_integral_elem_comp f g (h x) | lemma | ring_hom.is_integral_tower_top_of_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"is_integral",
"ring_hom.is_integral_elem_of_is_integral_elem_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_tower_top_of_is_integral {x : B} (h : is_integral R x) : is_integral A x | begin
rcases h with ⟨p, ⟨hp, hp'⟩⟩,
refine ⟨p.map (algebra_map R A), ⟨hp.map (algebra_map R A), _⟩⟩,
rw [is_scalar_tower.algebra_map_eq R A B, ← eval₂_map] at hp',
exact hp',
end | lemma | is_integral_tower_top_of_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"is_integral",
"is_scalar_tower.algebra_map_eq"
] | If `R → A → B` is an algebra tower,
then if the entire tower is an integral extension so is `A → B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom.is_integral_quotient_of_is_integral {I : ideal S} (hf : f.is_integral) :
(ideal.quotient_map I f le_rfl).is_integral | begin
rintros ⟨x⟩,
obtain ⟨p, ⟨p_monic, hpx⟩⟩ := hf x,
refine ⟨p.map (ideal.quotient.mk _), ⟨p_monic.map _, _⟩⟩,
simpa only [hom_eval₂, eval₂_map] using congr_arg (ideal.quotient.mk I) hpx
end | lemma | ring_hom.is_integral_quotient_of_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"ideal",
"ideal.quotient.mk",
"ideal.quotient_map",
"is_integral",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_quotient_of_is_integral {I : ideal A} (hRA : algebra.is_integral R A) :
algebra.is_integral (R ⧸ I.comap (algebra_map R A)) (A ⧸ I) | (algebra_map R A).is_integral_quotient_of_is_integral hRA | lemma | is_integral_quotient_of_is_integral | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra.is_integral",
"algebra_map",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_quotient_map_iff {I : ideal S} :
(ideal.quotient_map I f le_rfl).is_integral ↔
((ideal.quotient.mk I).comp f : R →+* S ⧸ I).is_integral | begin
let g := ideal.quotient.mk (I.comap f),
have := ideal.quotient_map_comp_mk le_rfl,
refine ⟨λ h, _, λ h, ring_hom.is_integral_tower_top_of_is_integral g _ (this ▸ h)⟩,
refine this ▸ ring_hom.is_integral_trans g (ideal.quotient_map I f le_rfl) _ h,
exact ring_hom.is_integral_of_surjective g ideal.quotient... | lemma | is_integral_quotient_map_iff | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"ideal",
"ideal.quotient.mk",
"ideal.quotient.mk_surjective",
"ideal.quotient_map",
"ideal.quotient_map_comp_mk",
"is_integral",
"le_rfl",
"ring_hom.is_integral_of_surjective",
"ring_hom.is_integral_tower_top_of_is_integral",
"ring_hom.is_integral_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_field_of_is_integral_of_is_field
{R S : Type*} [comm_ring R] [nontrivial R] [comm_ring S] [is_domain S]
[algebra R S] (H : algebra.is_integral R S) (hRS : function.injective (algebra_map R S))
(hS : is_field S) : is_field R | begin
refine ⟨⟨0, 1, zero_ne_one⟩, mul_comm, λ a ha, _⟩,
-- Let `a_inv` be the inverse of `algebra_map R S a`,
-- then we need to show that `a_inv` is of the form `algebra_map R S b`.
obtain ⟨a_inv, ha_inv⟩ := hS.mul_inv_cancel (λ h, ha (hRS (trans h (ring_hom.map_zero _).symm))),
-- Let `p : R[X]` be monic ... | lemma | is_field_of_is_integral_of_is_field | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra.is_integral",
"algebra_map",
"comm_ring",
"finset.range",
"finset.sum_mul",
"is_domain",
"is_field",
"le_tsub_of_add_le_left",
"mul_assoc",
"mul_comm",
"mul_pow",
"neg_mul",
"nontrivial",
"one_mul",
"one_ne_zero",
"one_pow",
"pow_add",
"pow_ne_zero",
"pow_su... | If the integral extension `R → S` is injective, and `S` is a field, then `R` is also a field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_field_of_is_integral_of_is_field'
{R S : Type*} [comm_ring R] [comm_ring S] [is_domain S] [algebra R S]
(H : algebra.is_integral R S) (hR : is_field R) :
is_field S | begin
letI := hR.to_field,
refine ⟨⟨0, 1, zero_ne_one⟩, mul_comm, λ x hx, _⟩,
let A := algebra.adjoin R ({x} : set S),
haveI : is_noetherian R A :=
is_noetherian_of_fg_of_noetherian A.to_submodule (fg_adjoin_singleton_of_integral x (H x)),
haveI : module.finite R A := module.is_noetherian.finite R A,
obta... | lemma | is_field_of_is_integral_of_is_field' | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra.adjoin",
"algebra.is_integral",
"comm_ring",
"fg_adjoin_singleton_of_integral",
"is_domain",
"is_field",
"is_noetherian",
"is_noetherian_of_fg_of_noetherian",
"linear_map.mul_left_injective",
"linear_map.surjective_of_injective",
"module.finite",
"module.is_noetherian.fin... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.is_integral.is_field_iff_is_field
{R S : Type*} [comm_ring R] [nontrivial R] [comm_ring S] [is_domain S] [algebra R S]
(H : algebra.is_integral R S) (hRS : function.injective (algebra_map R S)) :
is_field R ↔ is_field S | ⟨is_field_of_is_integral_of_is_field' H, is_field_of_is_integral_of_is_field H hRS⟩ | lemma | algebra.is_integral.is_field_iff_is_field | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"algebra.is_integral",
"algebra_map",
"comm_ring",
"is_domain",
"is_field",
"is_field_of_is_integral_of_is_field",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_closure_idem {R : Type*} {A : Type*} [comm_ring R] [comm_ring A] [algebra R A] :
integral_closure (integral_closure R A : set A) A = ⊥ | eq_bot_iff.2 $ λ x hx, algebra.mem_bot.2
⟨⟨x, @is_integral_trans _ _ _ _ _ _ _ _ (integral_closure R A).algebra
_ integral_closure.is_integral x hx⟩, rfl⟩ | theorem | integral_closure_idem | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra",
"comm_ring",
"integral_closure",
"integral_closure.is_integral",
"is_integral_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
roots_mem_integral_closure {f : R[X]} (hf : f.monic) {a : S}
(ha : a ∈ (f.map $ algebra_map R S).roots) : a ∈ integral_closure R S | ⟨f, hf, (eval₂_eq_eval_map _).trans $ (mem_roots $ (hf.map _).ne_zero).1 ha⟩ | theorem | roots_mem_integral_closure | ring_theory | src/ring_theory/integral_closure.lean | [
"data.polynomial.expand",
"linear_algebra.finite_dimensional",
"linear_algebra.matrix.charpoly.linear_map",
"ring_theory.adjoin.fg",
"ring_theory.finite_type",
"ring_theory.polynomial.scale_roots",
"ring_theory.polynomial.tower",
"ring_theory.tensor_product"
] | [
"algebra_map",
"integral_closure",
"ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_bijective_of_finite₀ {a : M} (ha : a ≠ 0) : bijective (λ b, a * b) | finite.injective_iff_bijective.1 $ mul_right_injective₀ ha | lemma | mul_right_bijective_of_finite₀ | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"mul_right_injective₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_bijective_of_finite₀ {a : M} (ha : a ≠ 0) : bijective (λ b, b * a) | finite.injective_iff_bijective.1 $ mul_left_injective₀ ha | lemma | mul_left_bijective_of_finite₀ | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"mul_left_injective₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype.group_with_zero_of_cancel (M : Type*) [cancel_monoid_with_zero M] [decidable_eq M]
[fintype M] [nontrivial M] : group_with_zero M | { inv := λ a, if h : a = 0 then 0 else fintype.bij_inv (mul_right_bijective_of_finite₀ h) 1,
mul_inv_cancel := λ a ha,
by { simp [has_inv.inv, dif_neg ha], exact fintype.right_inverse_bij_inv _ _ },
inv_zero := by { simp [has_inv.inv, dif_pos rfl] },
..‹nontrivial M›,
..‹cancel_monoid_with_zero M› } | def | fintype.group_with_zero_of_cancel | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"cancel_monoid_with_zero",
"fintype",
"fintype.bij_inv",
"fintype.right_inverse_bij_inv",
"group_with_zero",
"inv_zero",
"mul_inv_cancel",
"mul_right_bijective_of_finite₀",
"nontrivial"
] | Every finite nontrivial cancel_monoid_with_zero is a group_with_zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_eq_pow_of_mul_eq_pow_of_coprime {R : Type*} [comm_semiring R] [is_domain R]
[gcd_monoid R] [unique Rˣ] {a b c : R} {n : ℕ} (cp : is_coprime a b) (h : a * b = c ^ n) :
∃ d : R, a = d ^ n | begin
refine exists_eq_pow_of_mul_eq_pow (is_unit_of_dvd_one _ _) h,
obtain ⟨x, y, hxy⟩ := cp,
rw [← hxy],
exact dvd_add (dvd_mul_of_dvd_right (gcd_dvd_left _ _) _)
(dvd_mul_of_dvd_right (gcd_dvd_right _ _) _)
end | lemma | exists_eq_pow_of_mul_eq_pow_of_coprime | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"comm_semiring",
"dvd_add",
"dvd_mul_of_dvd_right",
"exists_eq_pow_of_mul_eq_pow",
"gcd_monoid",
"is_coprime",
"is_domain",
"is_unit_of_dvd_one",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.exists_eq_pow_of_mul_eq_pow_of_coprime {ι R : Type*} [comm_semiring R] [is_domain R]
[gcd_monoid R] [unique Rˣ] {n : ℕ} {c : R} {s : finset ι} {f : ι → R}
(h : ∀ i j ∈ s, i ≠ j → is_coprime (f i) (f j))
(hprod : ∏ i in s, f i = c ^ n) : ∀ i ∈ s, ∃ d : R, f i = d ^ n | begin
classical,
intros i hi,
rw [← insert_erase hi, prod_insert (not_mem_erase i s)] at hprod,
refine exists_eq_pow_of_mul_eq_pow_of_coprime
(is_coprime.prod_right (λ j hj, h i hi j (erase_subset i s hj) (λ hij, _))) hprod,
rw [hij] at hj,
exact (s.not_mem_erase _) hj
end | lemma | finset.exists_eq_pow_of_mul_eq_pow_of_coprime | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"comm_semiring",
"exists_eq_pow_of_mul_eq_pow_of_coprime",
"finset",
"gcd_monoid",
"is_coprime",
"is_coprime.prod_right",
"is_domain",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype.division_ring_of_is_domain (R : Type*) [ring R] [is_domain R] [decidable_eq R]
[fintype R] : division_ring R | { ..show group_with_zero R, from fintype.group_with_zero_of_cancel R,
..‹ring R› } | def | fintype.division_ring_of_is_domain | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"division_ring",
"fintype",
"fintype.group_with_zero_of_cancel",
"group_with_zero",
"is_domain",
"ring"
] | Every finite domain is a division ring.
TODO: Prove Wedderburn's little theorem,
which shows a finite domain is in fact commutative, hence a field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fintype.field_of_domain (R) [comm_ring R] [is_domain R] [decidable_eq R] [fintype R] :
field R | { .. fintype.group_with_zero_of_cancel R,
.. ‹comm_ring R› } | def | fintype.field_of_domain | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"comm_ring",
"field",
"fintype",
"fintype.group_with_zero_of_cancel",
"is_domain"
] | Every finite commutative domain is a field.
TODO: Prove Wedderburn's little theorem, which shows a finite domain is automatically commutative,
dropping one assumption from this theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite.is_field_of_domain (R) [comm_ring R] [is_domain R] [finite R] : is_field R | by { casesI nonempty_fintype R,
exact @field.to_is_field R (@@fintype.field_of_domain R _ _ (classical.dec_eq R) _) } | lemma | finite.is_field_of_domain | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"classical.dec_eq",
"comm_ring",
"field.to_is_field",
"finite",
"fintype.field_of_domain",
"is_domain",
"is_field",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_nth_roots_subgroup_units [fintype G] (f : G →* R) (hf : injective f) {n : ℕ} (hn : 0 < n)
(g₀ : G) :
({g ∈ univ | g ^ n = g₀} : finset G).card ≤ (nth_roots n (f g₀)).card | begin
haveI : decidable_eq R := classical.dec_eq _,
refine le_trans _ (nth_roots n (f g₀)).to_finset_card_le,
apply card_le_card_of_inj_on f,
{ intros g hg,
rw [sep_def, mem_filter] at hg,
rw [multiset.mem_to_finset, mem_nth_roots hn, ← f.map_pow, hg.2] },
{ intros, apply hf, assumption }
end | lemma | card_nth_roots_subgroup_units | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"classical.dec_eq",
"finset",
"fintype",
"multiset.mem_to_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cyclic_of_subgroup_is_domain [finite G] (f : G →* R) (hf : injective f) : is_cyclic G | begin
classical,
casesI nonempty_fintype G,
apply is_cyclic_of_card_pow_eq_one_le,
intros n hn,
convert (le_trans (card_nth_roots_subgroup_units f hf hn 1) (card_nth_roots n (f 1)))
end | lemma | is_cyclic_of_subgroup_is_domain | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"card_nth_roots_subgroup_units",
"finite",
"is_cyclic",
"is_cyclic_of_card_pow_eq_one_le",
"nonempty_fintype"
] | A finite subgroup of the unit group of an integral domain is cyclic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subgroup_units_cyclic : is_cyclic S | begin
refine is_cyclic_of_subgroup_is_domain ⟨(coe : S → R), _, _⟩
(units.ext.comp subtype.val_injective),
{ simp },
{ intros, simp },
end | instance | subgroup_units_cyclic | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"is_cyclic",
"is_cyclic_of_subgroup_is_domain",
"subtype.val_injective"
] | A finite subgroup of the units of an integral domain is cyclic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_eq_quo_add_rem_div (f : R[X]) {g : R[X]} (hg : g.monic) :
∃ q r : R[X], r.degree < g.degree ∧ (↑f : K) / ↑g = ↑q + ↑r / ↑g | begin
refine ⟨f /ₘ g, f %ₘ g, _, _⟩,
{ exact degree_mod_by_monic_lt _ hg },
{ have hg' : (↑g : K) ≠ 0 := by exact_mod_cast (monic.ne_zero hg),
field_simp [hg'],
norm_cast,
rw [add_comm, mul_comm, mod_by_monic_add_div f hg] },
end | lemma | polynomial.div_eq_quo_add_rem_div | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_fiber_eq_of_mem_range {H : Type*} [group H] [decidable_eq H]
(f : G →* H) {x y : H} (hx : x ∈ set.range f) (hy : y ∈ set.range f) :
(univ.filter $ λ g, f g = x).card = (univ.filter $ λ g, f g = y).card | begin
rcases hx with ⟨x, rfl⟩,
rcases hy with ⟨y, rfl⟩,
refine card_congr (λ g _, g * x⁻¹ * y) _ _ (λ g hg, ⟨g * y⁻¹ * x, _⟩),
{ simp only [mem_filter, one_mul, monoid_hom.map_mul, mem_univ, mul_right_inv,
eq_self_iff_true, monoid_hom.map_mul_inv, and_self, forall_true_iff] {contextual := tt} },
{ simp ... | lemma | card_fiber_eq_of_mem_range | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"exists_prop_of_true",
"forall_2_true_iff",
"forall_true_iff",
"group",
"imp_self",
"inv_mul_cancel_right",
"monoid_hom.map_mul",
"monoid_hom.map_mul_inv",
"mul_inv_cancel_right",
"mul_left_inj",
"mul_right_inv",
"one_mul",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_hom_units_eq_zero (f : G →* R) (hf : f ≠ 1) : ∑ g : G, f g = 0 | begin
classical,
obtain ⟨x, hx⟩ : ∃ x : monoid_hom.range f.to_hom_units,
∀ y : monoid_hom.range f.to_hom_units, y ∈ submonoid.powers x,
from is_cyclic.exists_monoid_generator,
have hx1 : x ≠ 1,
{ rintro rfl,
apply hf,
ext g,
rw [monoid_hom.one_apply],
cases hx ⟨f.to_hom_units g, g, rfl⟩ ... | lemma | sum_hom_units_eq_zero | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"card_fiber_eq_of_mem_range",
"coe_coe",
"congr_arg2",
"exists_prop_of_true",
"forall_true_iff",
"geom_sum_mul",
"is_cyclic.exists_monoid_generator",
"monoid_hom.coe_to_hom_units",
"monoid_hom.one_apply",
"monoid_hom.range",
"mul_left_inj'",
"one_pow",
"order_of",
"order_of_pos",
"pow_eq... | In an integral domain, a sum indexed by a nontrivial homomorphism from a finite group is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_hom_units (f : G →* R) [decidable (f = 1)] :
∑ g : G, f g = if f = 1 then fintype.card G else 0 | begin
split_ifs with h h,
{ simp [h, card_univ] },
{ exact sum_hom_units_eq_zero f h }
end | lemma | sum_hom_units | ring_theory | src/ring_theory/integral_domain.lean | [
"data.polynomial.ring_division",
"group_theory.specific_groups.cyclic",
"algebra.geom_sum"
] | [
"fintype.card",
"sum_hom_units_eq_zero"
] | In an integral domain, a sum indexed by a homomorphism from a finite group is zero,
unless the homomorphism is trivial, in which case the sum is equal to the cardinality of the group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_adjoin_root {R : Type u} (S : Type v) [comm_semiring R] [semiring S] [algebra R S]
(f : R[X]) : Type (max u v) | (map : R[X] →+* S)
(map_surjective : function.surjective map)
(ker_map : ring_hom.ker map = ideal.span {f})
(algebra_map_eq : algebra_map R S = map.comp polynomial.C) | structure | is_adjoin_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"ideal.span",
"polynomial.C",
"ring_hom.ker",
"semiring"
] | `is_adjoin_root S f` states that the ring `S` can be constructed by adjoining a specified root
of the polynomial `f : R[X]` to `R`.
Compare `power_basis R S`, which does not explicitly specify which polynomial we adjoin a root of
(in particular `f` does not need to be the minimal polynomial of the root we adjoin),
and... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_adjoin_root_monic {R : Type u} (S : Type v) [comm_semiring R] [semiring S]
[algebra R S] (f : R[X]) extends is_adjoin_root S f | (monic : monic f) | structure | is_adjoin_root_monic | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra",
"comm_semiring",
"is_adjoin_root",
"semiring"
] | `is_adjoin_root_monic S f` states that the ring `S` can be constructed by adjoining a specified
root of the monic polynomial `f : R[X]` to `R`.
As long as `f` is monic, there is a well-defined representation of elements of `S` as polynomials
in `R[X]` of degree lower than `deg f` (see `mod_by_monic_hom` and `coeff`). ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
root (h : is_adjoin_root S f) : S | h.map X | def | is_adjoin_root.root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | `(h : is_adjoin_root S f).root` is the root of `f` that can be adjoined to generate `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton (h : is_adjoin_root S f) [subsingleton R] : subsingleton S | h.map_surjective.subsingleton | lemma | is_adjoin_root.subsingleton | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply (h : is_adjoin_root S f) (x : R) :
algebra_map R S x = h.map (polynomial.C x) | by rw [h.algebra_map_eq, ring_hom.comp_apply] | lemma | is_adjoin_root.algebra_map_apply | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra_map",
"algebra_map_apply",
"is_adjoin_root",
"polynomial.C",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_ker_map (h : is_adjoin_root S f) {p} : p ∈ ring_hom.ker h.map ↔ f ∣ p | by rw [h.ker_map, ideal.mem_span_singleton] | lemma | is_adjoin_root.mem_ker_map | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"ideal.mem_span_singleton",
"is_adjoin_root",
"ring_hom.ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff (h : is_adjoin_root S f) {p} : h.map p = 0 ↔ f ∣ p | by rw [← h.mem_ker_map, ring_hom.mem_ker] | lemma | is_adjoin_root.map_eq_zero_iff | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_X (h : is_adjoin_root S f) : h.map X = h.root | rfl | lemma | is_adjoin_root.map_X | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_self (h : is_adjoin_root S f) : h.map f = 0 | h.map_eq_zero_iff.mpr dvd_rfl | lemma | is_adjoin_root.map_self | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"dvd_rfl",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_eq (h : is_adjoin_root S f) (p : R[X]) : aeval h.root p = h.map p | polynomial.induction_on p (λ x, by { rw [aeval_C, h.algebra_map_apply] })
(λ p q ihp ihq, by rw [alg_hom.map_add, ring_hom.map_add, ihp, ihq])
(λ n x ih, by { rw [alg_hom.map_mul, aeval_C, alg_hom.map_pow, aeval_X, ring_hom.map_mul,
← h.algebra_map_apply, ring_hom.map_pow, map_X] }) | lemma | is_adjoin_root.aeval_eq | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"alg_hom.map_add",
"alg_hom.map_mul",
"alg_hom.map_pow",
"ih",
"is_adjoin_root",
"polynomial.induction_on",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_root (h : is_adjoin_root S f) : aeval h.root f = 0 | by rw [aeval_eq, map_self] | lemma | is_adjoin_root.aeval_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr (h : is_adjoin_root S f) (x : S) : R[X] | (h.map_surjective x).some | def | is_adjoin_root.repr | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | Choose an arbitrary representative so that `h.map (h.repr x) = x`.
If `f` is monic, use `is_adjoin_root_monic.mod_by_monic_hom` for a unique choice of representative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_repr (h : is_adjoin_root S f) (x : S) : h.map (h.repr x) = x | (h.map_surjective x).some_spec | lemma | is_adjoin_root.map_repr | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
repr_zero_mem_span (h : is_adjoin_root S f) : h.repr 0 ∈ ideal.span ({f} : set R[X]) | by rw [← h.ker_map, ring_hom.mem_ker, h.map_repr] | lemma | is_adjoin_root.repr_zero_mem_span | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"ideal.span",
"is_adjoin_root",
"ring_hom.mem_ker"
] | `repr` preserves zero, up to multiples of `f` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
repr_add_sub_repr_add_repr_mem_span (h : is_adjoin_root S f) (x y : S) :
h.repr (x + y) - (h.repr x + h.repr y) ∈ ideal.span ({f} : set R[X]) | by rw [← h.ker_map, ring_hom.mem_ker, map_sub, h.map_repr, map_add, h.map_repr, h.map_repr,
sub_self] | lemma | is_adjoin_root.repr_add_sub_repr_add_repr_mem_span | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"ideal.span",
"is_adjoin_root",
"ring_hom.mem_ker"
] | `repr` preserves addition, up to multiples of `f` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_map (h h' : is_adjoin_root S f) (eq : ∀ x, h.map x = h'.map x) : h = h' | begin
cases h, cases h', congr,
exact ring_hom.ext eq
end | lemma | is_adjoin_root.ext_map | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root",
"ring_hom.ext"
] | Extensionality of the `is_adjoin_root` structure itself. See `is_adjoin_root_monic.ext_elem`
for extensionality of the ring elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext (h h' : is_adjoin_root S f) (eq : h.root = h'.root) : h = h' | h.ext_map h' (λ x, by rw [← h.aeval_eq, ← h'.aeval_eq, eq]) | lemma | is_adjoin_root.ext | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | Extensionality of the `is_adjoin_root` structure itself. See `is_adjoin_root_monic.ext_elem`
for extensionality of the ring elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval₂_repr_eq_eval₂_of_map_eq (h : is_adjoin_root S f) (z : S) (w : R[X])
(hzw : h.map w = z) :
(h.repr z).eval₂ i x = w.eval₂ i x | begin
rw [eq_comm, ← sub_eq_zero, ← h.map_repr z, ← map_sub, h.map_eq_zero_iff] at hzw,
obtain ⟨y, hy⟩ := hzw,
rw [← sub_eq_zero, ← eval₂_sub, hy, eval₂_mul, hx, zero_mul]
end | lemma | is_adjoin_root.eval₂_repr_eq_eval₂_of_map_eq | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root",
"zero_mul"
] | Auxiliary lemma for `is_adjoin_root.lift` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift (h : is_adjoin_root S f) : S →+* T | { to_fun := λ z, (h.repr z).eval₂ i x,
map_zero' := by rw [h.eval₂_repr_eq_eval₂_of_map_eq hx _ _ (map_zero _), eval₂_zero],
map_add' := λ z w, begin
rw [h.eval₂_repr_eq_eval₂_of_map_eq hx _ (h.repr z + h.repr w), eval₂_add],
{ rw [map_add, map_repr, map_repr] }
end,
map_one' := by rw [h.eval₂_repr_eq_e... | def | is_adjoin_root.lift | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root",
"lift",
"map_mul",
"map_one"
] | Lift a ring homomorphism `R →+* T` to `S →+* T` by specifying a root `x` of `f` in `T`,
where `S` is given by adjoining a root of `f` to `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_map (h : is_adjoin_root S f) (z : R[X]) :
h.lift i x hx (h.map z) = z.eval₂ i x | by rw [lift, ring_hom.coe_mk, h.eval₂_repr_eq_eval₂_of_map_eq hx _ _ rfl] | lemma | is_adjoin_root.lift_map | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root",
"lift",
"ring_hom.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_root (h : is_adjoin_root S f) :
h.lift i x hx h.root = x | by rw [← h.map_X, lift_map, eval₂_X] | lemma | is_adjoin_root.lift_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_algebra_map (h : is_adjoin_root S f) (a : R) :
h.lift i x hx (algebra_map R S a) = i a | by rw [h.algebra_map_apply, lift_map, eval₂_C] | lemma | is_adjoin_root.lift_algebra_map | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra_map",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_lift (h : is_adjoin_root S f) (g : S →+* T)
(hmap : ∀ a, g (algebra_map R S a) = i a) (hroot : g h.root = x) (a : S):
g a = h.lift i x hx a | begin
rw [← h.map_repr a, polynomial.as_sum_range_C_mul_X_pow (h.repr a)],
simp only [map_sum, map_mul, map_pow, h.map_X, hroot, ← h.algebra_map_apply, hmap, lift_root,
lift_algebra_map]
end | lemma | is_adjoin_root.apply_eq_lift | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra_map",
"is_adjoin_root",
"map_mul",
"map_pow",
"polynomial.as_sum_range_C_mul_X_pow"
] | Auxiliary lemma for `apply_eq_lift` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_lift (h : is_adjoin_root S f) (g : S →+* T)
(hmap : ∀ a, g (algebra_map R S a) = i a) (hroot : g h.root = x) :
g = h.lift i x hx | ring_hom.ext (h.apply_eq_lift hx g hmap hroot) | lemma | is_adjoin_root.eq_lift | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra_map",
"is_adjoin_root",
"ring_hom.ext"
] | Unicity of `lift`: a map that agrees on `R` and `h.root` agrees with `lift` everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_hom (h : is_adjoin_root S f) : S →ₐ[R] T | { commutes' := λ a, h.lift_algebra_map hx' a,
.. h.lift (algebra_map R T) x hx' } | def | is_adjoin_root.lift_hom | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra_map",
"is_adjoin_root"
] | Lift the algebra map `R → T` to `S →ₐ[R] T` by specifying a root `x` of `f` in `T`,
where `S` is given by adjoining a root of `f` to `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lift_hom (h : is_adjoin_root S f) :
(h.lift_hom x hx' : S →+* T) = h.lift (algebra_map R T) x hx' | rfl | lemma | is_adjoin_root.coe_lift_hom | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra_map",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_algebra_map_apply (h : is_adjoin_root S f) (z : S) :
h.lift (algebra_map R T) x hx' z = h.lift_hom x hx' z | rfl | lemma | is_adjoin_root.lift_algebra_map_apply | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra_map",
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_map (h : is_adjoin_root S f) (z : R[X]) :
h.lift_hom x hx' (h.map z) = aeval x z | by rw [← lift_algebra_map_apply, lift_map, aeval_def] | lemma | is_adjoin_root.lift_hom_map | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_root (h : is_adjoin_root S f) :
h.lift_hom x hx' h.root = x | by rw [← lift_algebra_map_apply, lift_root] | lemma | is_adjoin_root.lift_hom_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_lift_hom (h : is_adjoin_root S f) (g : S →ₐ[R] T) (hroot : g h.root = x) :
g = h.lift_hom x hx' | alg_hom.ext (h.apply_eq_lift hx' g g.commutes hroot) | lemma | is_adjoin_root.eq_lift_hom | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"alg_hom.ext",
"is_adjoin_root"
] | Unicity of `lift_hom`: a map that agrees on `h.root` agrees with `lift_hom` everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_adjoin_root : is_adjoin_root (adjoin_root f) f | { map := adjoin_root.mk f,
map_surjective := ideal.quotient.mk_surjective,
ker_map := begin
ext,
rw [ring_hom.mem_ker, ← @adjoin_root.mk_self _ _ f, adjoin_root.mk_eq_mk,
ideal.mem_span_singleton, ← dvd_add_left (dvd_refl f), sub_add_cancel]
end,
algebra_map_eq := adjoin_root.algebra_map_eq f } | def | adjoin_root.is_adjoin_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"adjoin_root",
"adjoin_root.algebra_map_eq",
"adjoin_root.mk",
"adjoin_root.mk_eq_mk",
"adjoin_root.mk_self",
"dvd_add_left",
"dvd_refl",
"ideal.mem_span_singleton",
"ideal.quotient.mk_surjective",
"is_adjoin_root",
"ring_hom.mem_ker"
] | `adjoin_root f` is indeed given by adjoining a root of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_adjoin_root_monic (hf : monic f) :
is_adjoin_root_monic (adjoin_root f) f | { monic := hf,
.. adjoin_root.is_adjoin_root f } | def | adjoin_root.is_adjoin_root_monic | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"adjoin_root",
"adjoin_root.is_adjoin_root",
"is_adjoin_root_monic"
] | `adjoin_root f` is indeed given by adjoining a root of `f`. If `f` is monic this is more
powerful than `adjoin_root.is_adjoin_root`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_adjoin_root_map_eq_mk :
(adjoin_root.is_adjoin_root f).map = adjoin_root.mk f | rfl | lemma | adjoin_root.is_adjoin_root_map_eq_mk | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"adjoin_root.is_adjoin_root",
"adjoin_root.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_adjoin_root_monic_map_eq_mk (hf : f.monic) :
(adjoin_root.is_adjoin_root_monic f hf).map = adjoin_root.mk f | rfl | lemma | adjoin_root.is_adjoin_root_monic_map_eq_mk | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"adjoin_root.is_adjoin_root_monic",
"adjoin_root.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_adjoin_root_root_eq_root :
(adjoin_root.is_adjoin_root f).root = adjoin_root.root f | by simp only [is_adjoin_root.root, adjoin_root.root, adjoin_root.is_adjoin_root_map_eq_mk] | lemma | adjoin_root.is_adjoin_root_root_eq_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"adjoin_root.is_adjoin_root",
"adjoin_root.is_adjoin_root_map_eq_mk",
"adjoin_root.root",
"is_adjoin_root.root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_adjoin_root_monic_root_eq_root (hf : monic f) :
(adjoin_root.is_adjoin_root_monic f hf).root = adjoin_root.root f | by simp only [is_adjoin_root.root, adjoin_root.root, adjoin_root.is_adjoin_root_monic_map_eq_mk] | lemma | adjoin_root.is_adjoin_root_monic_root_eq_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"adjoin_root.is_adjoin_root_monic",
"adjoin_root.is_adjoin_root_monic_map_eq_mk",
"adjoin_root.root",
"is_adjoin_root.root"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mod_by_monic (h : is_adjoin_root_monic S f) (g : R[X]) :
h.map (g %ₘ f) = h.map g | begin
rw [← ring_hom.sub_mem_ker_iff, mem_ker_map, mod_by_monic_eq_sub_mul_div _ h.monic,
sub_right_comm, sub_self, zero_sub, dvd_neg],
exact ⟨_, rfl⟩
end | lemma | is_adjoin_root_monic.map_mod_by_monic | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"dvd_neg",
"is_adjoin_root_monic",
"ring_hom.sub_mem_ker_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_by_monic_repr_map (h : is_adjoin_root_monic S f) (g : R[X]) :
h.repr (h.map g) %ₘ f = g %ₘ f | mod_by_monic_eq_of_dvd_sub h.monic $ by rw [← h.mem_ker_map, ring_hom.sub_mem_ker_iff, map_repr] | lemma | is_adjoin_root_monic.mod_by_monic_repr_map | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic",
"ring_hom.sub_mem_ker_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_by_monic_hom (h : is_adjoin_root_monic S f) : S →ₗ[R] R[X] | { to_fun := λ x, h.repr x %ₘ f,
map_add' := λ x y,
by conv_lhs { rw [← h.map_repr x, ← h.map_repr y, ← map_add, h.mod_by_monic_repr_map,
add_mod_by_monic] },
map_smul' := λ c x,
by rw [ring_hom.id_apply, ← h.map_repr x, algebra.smul_def, h.algebra_map_apply, ← map_mul,
h.mod... | def | is_adjoin_root_monic.mod_by_monic_hom | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"algebra.smul_def",
"is_adjoin_root_monic",
"map_mul",
"ring_hom.id_apply"
] | `is_adjoin_root.mod_by_monic_hom` sends the equivalence class of `f` mod `g` to `f %ₘ g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mod_by_monic_hom_map (h : is_adjoin_root_monic S f) (g : R[X]) :
h.mod_by_monic_hom (h.map g) = g %ₘ f | h.mod_by_monic_repr_map g | lemma | is_adjoin_root_monic.mod_by_monic_hom_map | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mod_by_monic_hom (h : is_adjoin_root_monic S f) (x : S) :
h.map (h.mod_by_monic_hom x) = x | by rw [mod_by_monic_hom, linear_map.coe_mk, map_mod_by_monic, map_repr] | lemma | is_adjoin_root_monic.map_mod_by_monic_hom | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic",
"linear_map.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_by_monic_hom_root_pow (h : is_adjoin_root_monic S f) {n : ℕ}
(hdeg : n < nat_degree f) :
h.mod_by_monic_hom (h.root ^ n) = X ^ n | begin
nontriviality R,
rwa [← h.map_X, ← map_pow, mod_by_monic_hom_map, mod_by_monic_eq_self_iff h.monic, degree_X_pow],
contrapose! hdeg,
simpa [nat_degree_le_iff_degree_le] using hdeg
end | lemma | is_adjoin_root_monic.mod_by_monic_hom_root_pow | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic",
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_by_monic_hom_root (h : is_adjoin_root_monic S f) (hdeg : 1 < nat_degree f) :
h.mod_by_monic_hom h.root = X | by simpa using mod_by_monic_hom_root_pow h hdeg | lemma | is_adjoin_root_monic.mod_by_monic_hom_root | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basis (h : is_adjoin_root_monic S f) : basis (fin (nat_degree f)) R S | basis.of_repr
{ to_fun := λ x, (h.mod_by_monic_hom x).to_finsupp.comap_domain coe (fin.coe_injective.inj_on _),
inv_fun := λ g, h.map (of_finsupp (g.map_domain coe)),
left_inv := λ x, begin
casesI subsingleton_or_nontrivial R,
{ haveI := h.subsingleton,
exact subsingleton.elim _ _ },
simp only,
... | def | is_adjoin_root_monic.basis | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"basis",
"fin.coe_injective",
"finset.mem_coe",
"finsupp.comap_domain_add_of_injective",
"finsupp.comap_domain_apply",
"finsupp.comap_domain_smul_of_injective",
"finsupp.map_domain_apply",
"finsupp.map_domain_comap_domain",
"finsupp.map_domain_notin_range",
"finsupp.mem_support_iff",
"inv_fun",
... | The basis on `S` generated by powers of `h.root`.
Auxiliary definition for `is_adjoin_root_monic.power_basis`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basis_apply (h : is_adjoin_root_monic S f) (i) : h.basis i = h.root ^ (i : ℕ) | basis.apply_eq_iff.mpr $
show (h.mod_by_monic_hom (h.to_is_adjoin_root.root ^ (i : ℕ))).to_finsupp
.comap_domain coe (fin.coe_injective.inj_on _) = finsupp.single _ _,
begin
ext j,
rw [finsupp.comap_domain_apply, mod_by_monic_hom_root_pow],
{ rw [X_pow_eq_monomial, to_finsupp_monomial, finsupp.single_apply_left... | lemma | is_adjoin_root_monic.basis_apply | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"fin.coe_injective",
"finsupp.comap_domain_apply",
"finsupp.single",
"finsupp.single_apply_left",
"is_adjoin_root_monic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deg_pos [nontrivial S] (h : is_adjoin_root_monic S f) : 0 < nat_degree f | begin
rcases h.basis.index_nonempty with ⟨⟨i, hi⟩⟩,
exact (nat.zero_le _).trans_lt hi
end | lemma | is_adjoin_root_monic.deg_pos | ring_theory | src/ring_theory/is_adjoin_root.lean | [
"data.polynomial.algebra_map",
"field_theory.minpoly.is_integrally_closed",
"ring_theory.power_basis"
] | [
"is_adjoin_root_monic",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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