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 |
|---|---|---|---|---|---|---|---|---|---|---|
comp_forget_augmented : simplicial_object.augmented (Type u) | simplicial_object.augment (classifying_space_universal_cover G ⋙ forget _) (terminal _)
(terminal.from _) $ λ i g h, subsingleton.elim _ _ | def | classifying_space_universal_cover.comp_forget_augmented | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"classifying_space_universal_cover"
] | The universal cover of the classifying space of `G` as a simplicial set, augmented by the map
from `fin 1 → G` to the terminal object in `Type u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extra_degeneracy_augmented_cech_nerve :
extra_degeneracy (arrow.mk $ terminal.from G).augmented_cech_nerve | augmented_cech_nerve.extra_degeneracy (arrow.mk $ terminal.from G)
⟨λ x, (1 : G), @subsingleton.elim _ (@unique.subsingleton _ (limits.unique_to_terminal _)) _ _⟩ | def | classifying_space_universal_cover.extra_degeneracy_augmented_cech_nerve | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [] | The augmented Čech nerve of the map from `fin 1 → G` to the terminal object in `Type u` has an
extra degeneracy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extra_degeneracy_comp_forget_augmented :
extra_degeneracy (comp_forget_augmented G) | begin
refine extra_degeneracy.of_iso (_ : (arrow.mk $ terminal.from G).augmented_cech_nerve ≅ _)
(extra_degeneracy_augmented_cech_nerve G),
exact comma.iso_mk (cech_nerve_terminal_from.iso G ≪≫
cech_nerve_terminal_from_iso_comp_forget G) (iso.refl _)
(by ext : 2; apply is_terminal.hom_ext terminal_is_te... | def | classifying_space_universal_cover.extra_degeneracy_comp_forget_augmented | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [] | The universal cover of the classifying space of `G` as a simplicial set, augmented by the map
from `fin 1 → G` to the terminal object in `Type u`, has an extra degeneracy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_forget_augmented.to_Module : simplicial_object.augmented (Module.{u} k) | ((simplicial_object.augmented.whiskering _ _).obj (Module.free k)).obj (comp_forget_augmented G) | def | classifying_space_universal_cover.comp_forget_augmented.to_Module | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Module.free"
] | The free functor `Type u ⥤ Module.{u} k` applied to the universal cover of the classifying
space of `G` as a simplicial set, augmented by the map from `fin 1 → G` to the terminal object
in `Type u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extra_degeneracy_comp_forget_augmented_to_Module :
extra_degeneracy (comp_forget_augmented.to_Module k G) | extra_degeneracy.map (extra_degeneracy_comp_forget_augmented G) (Module.free k) | def | classifying_space_universal_cover.extra_degeneracy_comp_forget_augmented_to_Module | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Module.free"
] | If we augment the universal cover of the classifying space of `G` as a simplicial set by the
map from `fin 1 → G` to the terminal object in `Type u`, then apply the free functor
`Type u ⥤ Module.{u} k`, the resulting augmented simplicial `k`-module has an extra degeneracy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_cohomology.resolution [monoid G] | (algebraic_topology.alternating_face_map_complex (Rep k G)).obj
(classifying_space_universal_cover G ⋙ (Rep.linearization k G).1.1) | def | group_cohomology.resolution | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Rep",
"Rep.linearization",
"algebraic_topology.alternating_face_map_complex",
"classifying_space_universal_cover",
"monoid"
] | The standard resolution of `k` as a trivial representation, defined as the alternating
face map complex of a simplicial `k`-linear `G`-representation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d (G : Type u) (n : ℕ) : ((fin (n + 1) → G) →₀ k) →ₗ[k] ((fin n → G) →₀ k) | finsupp.lift ((fin n → G) →₀ k) k (fin (n + 1) → G) (λ g, (@finset.univ (fin (n + 1)) _).sum
(λ p, finsupp.single (g ∘ p.succ_above) ((-1 : k) ^ (p : ℕ)))) | def | group_cohomology.resolution.d | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"finset.univ",
"finsupp.lift",
"finsupp.single"
] | The `k`-linear map underlying the differential in the standard resolution of `k` as a trivial
`k`-linear `G`-representation. It sends `(g₀, ..., gₙ) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_of {G : Type u} {n : ℕ} (c : fin (n + 1) → G) :
d k G n (finsupp.single c 1) = finset.univ.sum (λ p : fin (n + 1), finsupp.single
(c ∘ p.succ_above) ((-1 : k) ^ (p : ℕ))) | by simp [d] | lemma | group_cohomology.resolution.d_of | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"finsupp.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
X_iso (n : ℕ) :
(group_cohomology.resolution k G).X n ≅ Rep.of_mul_action k G (fin (n + 1) → G) | iso.refl _ | def | group_cohomology.resolution.X_iso | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Rep.of_mul_action",
"group_cohomology.resolution"
] | The `n`th object of the standard resolution of `k` is definitionally isomorphic to `k[Gⁿ⁺¹]`
equipped with the representation induced by the diagonal action of `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
X_projective (G : Type u) [group G] (n : ℕ) :
projective ((group_cohomology.resolution k G).X n) | Rep.equivalence_Module_monoid_algebra.to_adjunction.projective_of_map_projective _ $
@Module.projective_of_free.{u} _ _ (Module.of (monoid_algebra k G)
(representation.of_mul_action k G (fin (n + 1) → G)).as_module) _ (of_mul_action_basis k G n) | lemma | group_cohomology.resolution.X_projective | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Module.of",
"group",
"group_cohomology.resolution",
"monoid_algebra",
"representation.of_mul_action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
d_eq (n : ℕ) :
((group_cohomology.resolution k G).d (n + 1) n).hom = d k G (n + 1) | begin
ext x y,
dsimp [group_cohomology.resolution],
simpa [←@int_cast_smul k, simplicial_object.δ],
end | theorem | group_cohomology.resolution.d_eq | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"group_cohomology.resolution",
"int_cast_smul"
] | Simpler expression for the differential in the standard resolution of `k` as a
`G`-representation. It sends `(g₀, ..., gₙ₊₁) ↦ ∑ (-1)ⁱ • (g₀, ..., ĝᵢ, ..., gₙ₊₁)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_to_Module | ((forget₂ (Rep k G) (Module.{u} k)).map_homological_complex _).obj
(group_cohomology.resolution k G) | def | group_cohomology.resolution.forget₂_to_Module | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Rep",
"group_cohomology.resolution"
] | The standard resolution of `k` as a trivial representation as a complex of `k`-modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_forget_augmented_iso : (alternating_face_map_complex.obj
(simplicial_object.augmented.drop.obj (comp_forget_augmented.to_Module k G))) ≅
group_cohomology.resolution.forget₂_to_Module k G | eq_to_iso (functor.congr_obj (map_alternating_face_map_complex (forget₂ (Rep k G)
(Module.{u} k))).symm (classifying_space_universal_cover G ⋙ (Rep.linearization k G).1.1)) | def | group_cohomology.resolution.comp_forget_augmented_iso | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Rep",
"Rep.linearization",
"classifying_space_universal_cover",
"group_cohomology.resolution.forget₂_to_Module"
] | If we apply the free functor `Type u ⥤ Module.{u} k` to the universal cover of the classifying
space of `G` as a simplicial set, then take the alternating face map complex, the result is
isomorphic to the standard resolution of the trivial `G`-representation `k` as a complex of
`k`-modules. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_to_Module_homotopy_equiv : homotopy_equiv
(group_cohomology.resolution.forget₂_to_Module k G)
((chain_complex.single₀ (Module k)).obj
((forget₂ (Rep k G) _).obj $ Rep.trivial k G k)) | (homotopy_equiv.of_iso (comp_forget_augmented_iso k G).symm).trans $
(simplicial_object.augmented.extra_degeneracy.homotopy_equiv
(extra_degeneracy_comp_forget_augmented_to_Module k G)).trans
(homotopy_equiv.of_iso $ (chain_complex.single₀ (Module.{u} k)).map_iso
(@finsupp.linear_equiv.finsupp_unique k k _ ... | def | group_cohomology.resolution.forget₂_to_Module_homotopy_equiv | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Module",
"Rep",
"Rep.trivial",
"chain_complex.single₀",
"finsupp.linear_equiv.finsupp_unique",
"group_cohomology.resolution.forget₂_to_Module",
"homotopy_equiv",
"homotopy_equiv.of_iso",
"simplicial_object.augmented.extra_degeneracy.homotopy_equiv"
] | As a complex of `k`-modules, the standard resolution of the trivial `G`-representation `k` is
homotopy equivalent to the complex which is `k` at 0 and 0 elsewhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ε : Rep.of_mul_action k G (fin 1 → G) ⟶ Rep.trivial k G k | { hom := finsupp.total _ _ _ (λ f, (1 : k)),
comm' := λ g,
begin
ext,
show finsupp.total (fin 1 → G) k k (λ f, (1 : k))
(finsupp.map_domain _ (finsupp.single _ _)) = finsupp.total _ _ _ _ (finsupp.single _ _),
simp only [finsupp.map_domain_single, finsupp.total_single],
end } | def | group_cohomology.resolution.ε | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Rep.of_mul_action",
"Rep.trivial",
"finsupp.map_domain",
"finsupp.map_domain_single",
"finsupp.single",
"finsupp.total",
"finsupp.total_single"
] | The hom of `k`-linear `G`-representations `k[G¹] → k` sending `∑ nᵢgᵢ ↦ ∑ nᵢ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget₂_to_Module_homotopy_equiv_f_0_eq :
(forget₂_to_Module_homotopy_equiv k G).1.f 0 =
(forget₂ (Rep k G) _).map (ε k G) | begin
show (homotopy_equiv.hom _ ≫ (homotopy_equiv.hom _ ≫ homotopy_equiv.hom _)).f 0 = _,
simp only [homological_complex.comp_f],
convert category.id_comp _,
{ dunfold homotopy_equiv.of_iso comp_forget_augmented_iso map_alternating_face_map_complex,
simp only [iso.symm_hom, eq_to_iso.inv, homological_compl... | lemma | group_cohomology.resolution.forget₂_to_Module_homotopy_equiv_f_0_eq | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Module.free",
"Rep",
"classifying_space_universal_cover",
"finsupp.lmap_domain_total",
"finsupp.single_eq_same",
"finsupp.total",
"finsupp.total_single",
"homological_complex.comp_f",
"homological_complex.eq_to_hom_f",
"homotopy_equiv.of_iso",
"linear_map.id",
"linear_map.id_comp",
"one_smu... | The homotopy equivalence of complexes of `k`-modules between the standard resolution of `k` as
a trivial `G`-representation, and the complex which is `k` at 0 and 0 everywhere else, acts as
`∑ nᵢgᵢ ↦ ∑ nᵢ : k[G¹] → k` at 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
d_comp_ε : (group_cohomology.resolution k G).d 1 0 ≫ ε k G = 0 | begin
ext1,
refine linear_map.ext (λ x, _),
have : (forget₂_to_Module k G).d 1 0 ≫ (forget₂ (Rep k G) (Module.{u} k)).map (ε k G) = 0,
by rw [←forget₂_to_Module_homotopy_equiv_f_0_eq,
←(forget₂_to_Module_homotopy_equiv k G).1.2 1 0 rfl]; exact comp_zero,
exact linear_map.ext_iff.1 this _,
end | lemma | group_cohomology.resolution.d_comp_ε | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Rep",
"group_cohomology.resolution",
"linear_map.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ε_to_single₀ : group_cohomology.resolution k G ⟶ (chain_complex.single₀ _).obj
(Rep.trivial k G k) | ((group_cohomology.resolution k G).to_single₀_equiv _).symm ⟨ε k G, d_comp_ε k G⟩ | def | group_cohomology.resolution.ε_to_single₀ | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Rep.trivial",
"chain_complex.single₀",
"group_cohomology.resolution"
] | The chain map from the standard resolution of `k` to `k[0]` given by `∑ nᵢgᵢ ↦ ∑ nᵢ` in
degree zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ε_to_single₀_comp_eq : ((forget₂ _ (Module.{u} k)).map_homological_complex _).map
(ε_to_single₀ k G) ≫ ((chain_complex.single₀_map_homological_complex _).hom.app _) =
(forget₂_to_Module_homotopy_equiv k G).hom | begin
refine chain_complex.to_single₀_ext _ _ _,
dsimp,
rw category.comp_id,
exact (forget₂_to_Module_homotopy_equiv_f_0_eq k G).symm,
end | lemma | group_cohomology.resolution.ε_to_single₀_comp_eq | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"chain_complex.single₀_map_homological_complex",
"chain_complex.to_single₀_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_iso_of_forget₂_ε_to_single₀ :
quasi_iso (((forget₂ _ (Module.{u} k)).map_homological_complex _).map (ε_to_single₀ k G)) | begin
have h : quasi_iso (forget₂_to_Module_homotopy_equiv k G).hom := homotopy_equiv.to_quasi_iso _,
rw ← ε_to_single₀_comp_eq k G at h,
haveI := h,
exact quasi_iso_of_comp_right _ (((chain_complex.single₀_map_homological_complex _).hom.app _)),
end | lemma | group_cohomology.resolution.quasi_iso_of_forget₂_ε_to_single₀ | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"chain_complex.single₀_map_homological_complex",
"homotopy_equiv.to_quasi_iso",
"quasi_iso",
"quasi_iso_of_comp_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_cohomology.ProjectiveResolution :
ProjectiveResolution (Rep.trivial k G k) | (ε_to_single₀ k G).to_single₀_ProjectiveResolution (X_projective k G) | def | group_cohomology.ProjectiveResolution | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Rep.trivial"
] | The standard projective resolution of `k` as a trivial `k`-linear `G`-representation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group_cohomology.Ext_iso (V : Rep k G) (n : ℕ) :
((Ext k (Rep k G) n).obj (opposite.op $ Rep.trivial k G k)).obj V ≅
(((((linear_yoneda k (Rep k G)).obj V).right_op.map_homological_complex _).obj
(group_cohomology.resolution k G)).homology n).unop | by let := (((linear_yoneda k (Rep k G)).obj V).right_op.left_derived_obj_iso
n (group_cohomology.ProjectiveResolution k G)).unop.symm; exact this | def | group_cohomology.Ext_iso | representation_theory.group_cohomology | src/representation_theory/group_cohomology/resolution.lean | [
"algebra.category.Module.projective",
"algebraic_topology.extra_degeneracy",
"category_theory.abelian.ext",
"representation_theory.Rep"
] | [
"Ext",
"Rep",
"Rep.trivial",
"group_cohomology.ProjectiveResolution",
"group_cohomology.resolution",
"homology",
"opposite.op"
] | Given a `k`-linear `G`-representation `V`, `Extⁿ(k, V)` (where `k` is a trivial `k`-linear
`G`-representation) is isomorphic to the `n`th cohomology group of `Hom(P, V)`, where `P` is the
standard resolution of `k` called `group_cohomology.resolution k G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adjoin_root [comm_ring R] (f : R[X]) : Type u | polynomial R ⧸ (span {f} : ideal R[X]) | def | adjoin_root | 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"
] | [
"comm_ring",
"ideal",
"polynomial"
] | Adjoin a root of a polynomial `f` to a commutative ring `R`. We define the new ring
as the quotient of `R[X]` by the principal ideal generated by `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nontrivial [is_domain R] (h : degree f ≠ 0) : nontrivial (adjoin_root f) | ideal.quotient.nontrivial
begin
simp_rw [ne.def, span_singleton_eq_top, polynomial.is_unit_iff, not_exists, not_and],
rintro x hx rfl,
exact h (degree_C hx.ne_zero),
end | lemma | adjoin_root.nontrivial | 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.quotient.nontrivial",
"is_domain",
"nontrivial",
"not_and",
"not_exists",
"polynomial.is_unit_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk : R[X] →+* adjoin_root f | ideal.quotient.mk _ | def | adjoin_root.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",
"ideal.quotient.mk"
] | Ring homomorphism from `R[x]` to `adjoin_root f` sending `X` to the `root`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induction_on {C : adjoin_root f → Prop} (x : adjoin_root f)
(ih : ∀ p : R[X], C (mk f p)) : C x | quotient.induction_on' x ih | theorem | adjoin_root.induction_on | 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",
"ih",
"quotient.induction_on'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of : R →+* adjoin_root f | (mk f).comp C | def | adjoin_root.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"
] | [
"adjoin_root"
] | Embedding of the original ring `R` into `adjoin_root f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_mk [distrib_smul S R] [is_scalar_tower S R R] (a : S) (x : R[X]) :
a • mk f x = mk f (a • x) | rfl | lemma | adjoin_root.smul_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"
] | [
"distrib_smul",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_of [distrib_smul S R] [is_scalar_tower S R R] (a : S) (x : R) :
a • of f x = of f (a • x) | by rw [of, ring_hom.comp_apply, ring_hom.comp_apply, smul_mk, smul_C] | lemma | adjoin_root.smul_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"
] | [
"distrib_smul",
"is_scalar_tower",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_right [distrib_smul S R] [is_scalar_tower S R R] :
is_scalar_tower S (adjoin_root f) (adjoin_root f) | ideal.quotient.is_scalar_tower_right | instance | adjoin_root.is_scalar_tower_right | 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",
"distrib_smul",
"ideal.quotient.is_scalar_tower_right",
"is_scalar_tower"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq : algebra_map R (adjoin_root f) = of f | rfl | lemma | adjoin_root.algebra_map_eq | 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",
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eq' [comm_semiring S] [algebra S R] :
algebra_map S (adjoin_root f) = (of f).comp (algebra_map S R) | rfl | lemma | adjoin_root.algebra_map_eq' | 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",
"algebra",
"algebra_map",
"comm_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_type : algebra.finite_type R (adjoin_root f) | (algebra.finite_type.polynomial R).of_surjective _ (ideal.quotient.mkₐ_surjective R _) | lemma | adjoin_root.finite_type | 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",
"algebra.finite_type",
"algebra.finite_type.polynomial",
"ideal.quotient.mkₐ_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_presentation : algebra.finite_presentation R (adjoin_root f) | (algebra.finite_presentation.polynomial R).quotient (submodule.fg_span_singleton f) | lemma | adjoin_root.finite_presentation | 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",
"algebra.finite_presentation",
"algebra.finite_presentation.polynomial",
"submodule.fg_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
root : adjoin_root f | mk f X | def | adjoin_root.root | 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"
] | The adjoined root. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coe_t : has_coe_t R (adjoin_root f) | ⟨of f⟩ | instance | adjoin_root.has_coe_t | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom_ext [semiring S] [algebra R S] {g₁ g₂ : adjoin_root f →ₐ[R] S}
(h : g₁ (root f) = g₂ (root f)) : g₁ = g₂ | ideal.quotient.alg_hom_ext R $ polynomial.alg_hom_ext h | lemma | adjoin_root.alg_hom_ext | 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",
"algebra",
"ideal.quotient.alg_hom_ext",
"polynomial.alg_hom_ext",
"semiring"
] | Two `R`-`alg_hom` from `adjoin_root f` to the same `R`-algebra are the same iff
they agree on `root f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_eq_mk {g h : R[X]} : mk f g = mk f h ↔ f ∣ g - h | ideal.quotient.eq.trans ideal.mem_span_singleton | lemma | adjoin_root.mk_eq_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.mem_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_zero {g : R[X]} : mk f g = 0 ↔ f ∣ g | mk_eq_mk.trans $ by rw sub_zero | lemma | adjoin_root.mk_eq_zero | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_self : mk f f = 0 | quotient.sound' $ quotient_add_group.left_rel_apply.mpr (mem_span_singleton.2 $ by simp) | lemma | adjoin_root.mk_self | 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"
] | [
"quotient.sound'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_C (x : R) : mk f (C x) = x | rfl | lemma | adjoin_root.mk_C | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_X : mk f X = root f | rfl | lemma | adjoin_root.mk_X | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ne_zero_of_degree_lt (hf : monic f)
{g : R[X]} (h0 : g ≠ 0) (hd : degree g < degree f) : mk f g ≠ 0 | mk_eq_zero.not.2 $ hf.not_dvd_of_degree_lt h0 hd | lemma | adjoin_root.mk_ne_zero_of_degree_lt | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ne_zero_of_nat_degree_lt (hf : monic f)
{g : R[X]} (h0 : g ≠ 0) (hd : nat_degree g < nat_degree f) : mk f g ≠ 0 | mk_eq_zero.not.2 $ hf.not_dvd_of_nat_degree_lt h0 hd | lemma | adjoin_root.mk_ne_zero_of_nat_degree_lt | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_eq (p : R[X]) : aeval (root f) p = mk f p | polynomial.induction_on p (λ x, by { rw aeval_C, refl })
(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, mk_C, ring_hom.map_pow, mk_X], refl }) | lemma | adjoin_root.aeval_eq | 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"
] | [
"alg_hom.map_add",
"alg_hom.map_mul",
"alg_hom.map_pow",
"ih",
"polynomial.induction_on",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_root_eq_top : algebra.adjoin R ({root f} : set (adjoin_root f)) = ⊤ | algebra.eq_top_iff.2 $ λ x, induction_on f x $ λ p,
(algebra.adjoin_singleton_eq_range_aeval R (root f)).symm ▸ ⟨p, aeval_eq p⟩ | theorem | adjoin_root.adjoin_root_eq_top | 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",
"algebra.adjoin",
"algebra.adjoin_singleton_eq_range_aeval"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval₂_root (f : R[X]) : f.eval₂ (of f) (root f) = 0 | by rw [← algebra_map_eq, ← aeval_def, aeval_eq, mk_self] | lemma | adjoin_root.eval₂_root | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_root_root (f : R[X]) : is_root (f.map (of f)) (root f) | by rw [is_root, eval_map, eval₂_root] | lemma | adjoin_root.is_root_root | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_algebraic_root (hf : f ≠ 0) : is_algebraic R (root f) | ⟨f, hf, eval₂_root f⟩ | lemma | adjoin_root.is_algebraic_root | 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"
] | [
"is_algebraic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of.injective_of_degree_ne_zero [is_domain R] (hf : f.degree ≠ 0) :
function.injective (adjoin_root.of f) | begin
rw injective_iff_map_eq_zero,
intros p hp,
rw [adjoin_root.of, ring_hom.comp_apply, adjoin_root.mk_eq_zero] at hp,
by_cases h : f = 0,
{ exact C_eq_zero.mp (eq_zero_of_zero_dvd (by rwa h at hp)) },
{ contrapose! hf with h_contra,
rw ← degree_C h_contra,
apply le_antisymm (degree_le_of_dvd hp (... | lemma | adjoin_root.of.injective_of_degree_ne_zero | 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_eq_zero",
"adjoin_root.of",
"eq_zero_of_zero_dvd",
"is_domain",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (i : R →+* S) (x : S) (h : f.eval₂ i x = 0) : (adjoin_root f) →+* S | begin
apply ideal.quotient.lift _ (eval₂_ring_hom i x),
intros g H,
rcases mem_span_singleton.1 H with ⟨y, hy⟩,
rw [hy, ring_hom.map_mul, coe_eval₂_ring_hom, h, zero_mul]
end | def | adjoin_root.lift | 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.quotient.lift",
"lift",
"ring_hom.map_mul",
"zero_mul"
] | Lift a ring homomorphism `i : R →+* S` to `adjoin_root f →+* S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk (g : R[X]) : lift i a h (mk f g) = g.eval₂ i a | ideal.quotient.lift_mk _ _ _ | lemma | adjoin_root.lift_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.lift_mk",
"lift",
"lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_root : lift i a h (root f) = a | by rw [root, lift_mk, eval₂_X] | lemma | adjoin_root.lift_root | 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"
] | [
"lift",
"lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of {x : R} : lift i a h x = i x | by rw [← mk_C x, lift_mk, eval₂_C] | lemma | adjoin_root.lift_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"
] | [
"lift",
"lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_of : (lift i a h).comp (of f) = i | ring_hom.ext $ λ _, @lift_of _ _ _ _ _ _ _ h _ | lemma | adjoin_root.lift_comp_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"
] | [
"lift",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom (x : S) (hfx : aeval x f = 0) : adjoin_root f →ₐ[R] S | { commutes' := λ r, show lift _ _ hfx r = _, from lift_of hfx,
.. lift (algebra_map R S) x hfx } | def | adjoin_root.lift_hom | 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",
"algebra_map",
"lift"
] | Produce an algebra homomorphism `adjoin_root f →ₐ[R] S` sending `root f` to
a root of `f` in `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_lift_hom (x : S) (hfx : aeval x f = 0) :
(lift_hom f x hfx : adjoin_root f →+* S) = lift (algebra_map R S) x hfx | rfl | lemma | adjoin_root.coe_lift_hom | 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",
"algebra_map",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_alg_hom_eq_zero (ϕ : adjoin_root f →ₐ[R] S) : aeval (ϕ (root f)) f = 0 | begin
have h : ϕ.to_ring_hom.comp (of f) = algebra_map R S := ring_hom.ext_iff.mpr (ϕ.commutes),
rw [aeval_def, ←h, ←ring_hom.map_zero ϕ.to_ring_hom, ←eval₂_root f, hom_eval₂],
refl,
end | lemma | adjoin_root.aeval_alg_hom_eq_zero | 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",
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_eq_alg_hom (f : R[X]) (ϕ : adjoin_root f →ₐ[R] S) :
lift_hom f (ϕ (root f)) (aeval_alg_hom_eq_zero f ϕ) = ϕ | begin
suffices : ϕ.equalizer (lift_hom f (ϕ (root f)) (aeval_alg_hom_eq_zero f ϕ)) = ⊤,
{ exact (alg_hom.ext (λ x, (set_like.ext_iff.mp (this) x).mpr algebra.mem_top)).symm },
rw [eq_top_iff, ←adjoin_root_eq_top, algebra.adjoin_le_iff, set.singleton_subset_iff],
exact (@lift_root _ _ _ _ _ _ _ (aeval_alg_hom_eq... | lemma | adjoin_root.lift_hom_eq_alg_hom | 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",
"alg_hom.ext",
"algebra.adjoin_le_iff",
"algebra.mem_top",
"eq_top_iff",
"set.singleton_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_mk {g : R[X]} : lift_hom f a hfx (mk f g) = aeval a g | lift_mk hfx g | lemma | adjoin_root.lift_hom_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"
] | [
"lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_root : lift_hom f a hfx (root f) = a | lift_root hfx | lemma | adjoin_root.lift_hom_root | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_hom_of {x : R} : lift_hom f a hfx (of f x) = algebra_map _ _ x | lift_of hfx | lemma | adjoin_root.lift_hom_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"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
root_is_inv (r : R) : of _ r * root (C r * X - 1) = 1 | by convert sub_eq_zero.1 ((eval₂_sub _).symm.trans $ eval₂_root $ C r * X - 1);
simp only [eval₂_mul, eval₂_C, eval₂_X, eval₂_one] | lemma | adjoin_root.root_is_inv | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom_subsingleton {S : Type*} [comm_ring S] [algebra R S] {r : R} :
subsingleton (adjoin_root (C r * X - 1) →ₐ[R] S) | ⟨λ f g, alg_hom_ext (@inv_unique _ _ (algebra_map R S r) _ _
(by rw [← f.commutes, ← f.map_mul, algebra_map_eq, root_is_inv, map_one])
(by rw [← g.commutes, ← g.map_mul, algebra_map_eq, root_is_inv, map_one]))⟩ | lemma | adjoin_root.alg_hom_subsingleton | 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",
"algebra",
"algebra_map",
"comm_ring",
"inv_unique",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_domain_of_prime (hf : prime f) : is_domain (adjoin_root f) | (ideal.quotient.is_domain_iff_prime (span {f} : ideal R[X])).mpr $
(ideal.span_singleton_prime hf.ne_zero).mpr hf | theorem | adjoin_root.is_domain_of_prime | 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",
"ideal.quotient.is_domain_iff_prime",
"ideal.span_singleton_prime",
"is_domain",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
no_zero_smul_divisors_of_prime_of_degree_ne_zero [is_domain R] (hf : prime f)
(hf' : f.degree ≠ 0) : no_zero_smul_divisors R (adjoin_root f) | begin
haveI := is_domain_of_prime hf,
exact no_zero_smul_divisors.iff_algebra_map_injective.mpr (of.injective_of_degree_ne_zero hf')
end | theorem | adjoin_root.no_zero_smul_divisors_of_prime_of_degree_ne_zero | 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",
"is_domain",
"no_zero_smul_divisors",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_maximal_of_irreducible [fact (irreducible f)] : (span {f}).is_maximal | principal_ideal_ring.is_maximal_of_irreducible $ fact.out _ | instance | adjoin_root.span_maximal_of_irreducible | 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"
] | [
"fact",
"irreducible",
"principal_ideal_ring.is_maximal_of_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
field [fact (irreducible f)] : field (adjoin_root f) | { rat_cast := λ a, of f (a : K),
rat_cast_mk := λ a b h1 h2,
begin
letI : group_with_zero (adjoin_root f) := ideal.quotient.group_with_zero _,
rw [rat.cast_mk', _root_.map_mul, _root_.map_int_cast, map_inv₀, map_nat_cast],
end,
qsmul := (•),
qsmul_eq_mul' := λ a x, adjoin_root.induction_on _ x (λ p,
... | instance | adjoin_root.field | 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.induction_on",
"fact",
"field",
"group_with_zero",
"ideal",
"ideal.quotient.group_with_zero",
"irreducible",
"map_inv₀",
"map_mul",
"map_nat_cast",
"polynomial.rat_smul_eq_C_mul",
"rat.cast_mk'",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective (h : degree f ≠ 0) : function.injective (coe : K → adjoin_root f) | have _ := adjoin_root.nontrivial f h, by exactI (of f).injective | lemma | adjoin_root.coe_injective | 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.nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective' [fact (irreducible f)] : function.injective (coe : K → adjoin_root f) | (of f).injective | lemma | adjoin_root.coe_injective' | 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",
"fact",
"irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_root_cancel [fact (irreducible f)] :
((X - C (root f)) * (f.map (of f) / (X - C (root f)))) = f.map (of f) | mul_div_eq_iff_is_root.2 $ is_root_root _ | lemma | adjoin_root.mul_div_root_cancel | 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"
] | [
"fact",
"irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_integral_root' (hg : g.monic) : is_integral R (root g) | ⟨g, hg, eval₂_root g⟩ | lemma | adjoin_root.is_integral_root' | 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"
] | [
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_by_monic_hom (hg : g.monic) :
adjoin_root g →ₗ[R] R[X] | (submodule.liftq _ (polynomial.mod_by_monic_hom g)
(λ f (hf : f ∈ (ideal.span {g}).restrict_scalars R),
(mem_ker_mod_by_monic hg).mpr (ideal.mem_span_singleton.mp hf))).comp $
(submodule.quotient.restrict_scalars_equiv R (ideal.span {g} : ideal R[X]))
.symm.to_linear_map | def | adjoin_root.mod_by_monic_hom | 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",
"ideal.span",
"polynomial.mod_by_monic_hom",
"restrict_scalars",
"submodule.liftq",
"submodule.quotient.restrict_scalars_equiv"
] | `adjoin_root.mod_by_monic_hom` sends the equivalence class of `f` mod `g` to `f %ₘ g`.
This is a well-defined right inverse to `adjoin_root.mk`, see `adjoin_root.mk_left_inverse`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mod_by_monic_hom_mk (hg : g.monic) (f : R[X]) :
mod_by_monic_hom hg (mk g f) = f %ₘ g | rfl | lemma | adjoin_root.mod_by_monic_hom_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_left_inverse (hg : g.monic) :
function.left_inverse (mk g) (mod_by_monic_hom hg) | λ f, induction_on g f $ λ f, begin
rw [mod_by_monic_hom_mk hg, mk_eq_mk, mod_by_monic_eq_sub_mul_div _ hg,
sub_sub_cancel_left, dvd_neg],
apply dvd_mul_right
end | lemma | adjoin_root.mk_left_inverse | 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"
] | [
"dvd_mul_right",
"dvd_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_surjective (hg : g.monic) : function.surjective (mk g) | (mk_left_inverse hg).surjective | lemma | adjoin_root.mk_surjective | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_basis_aux' (hg : g.monic) :
basis (fin g.nat_degree) R (adjoin_root g) | basis.of_equiv_fun
{ to_fun := λ f i, (mod_by_monic_hom hg f).coeff i,
inv_fun := λ c, mk g $ ∑ (i : fin g.nat_degree), monomial i (c i),
map_add' := λ f₁ f₂, funext $ λ i,
by simp only [(mod_by_monic_hom hg).map_add, coeff_add, pi.add_apply],
map_smul' := λ f₁ f₂, funext $ λ i,
by simp only [(mod_by_moni... | def | adjoin_root.power_basis_aux' | 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",
"basis",
"basis.of_equiv_fun",
"dvd_mul_right",
"fin.coe_eq_coe",
"finset.mem_univ",
"inv_fun",
"pi.smul_apply",
"ring_hom.id_apply"
] | The elements `1, root g, ..., root g ^ (d - 1)` form a basis for `adjoin_root g`,
where `g` is a monic polynomial of degree `d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_basis_aux'_repr_symm_apply (hg : g.monic) (c : fin g.nat_degree →₀ R) :
(power_basis_aux' hg).repr.symm c = mk g (∑ (i : fin _), monomial i (c i)) | rfl | lemma | adjoin_root.power_basis_aux'_repr_symm_apply | 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"
] | [] | This lemma could be autogenerated by `@[simps]` but unfortunately that would require
unfolding that causes a timeout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_basis_aux'_repr_apply_to_fun (hg : g.monic) (f : adjoin_root g)
(i : fin g.nat_degree) :
(power_basis_aux' hg).repr f i = (mod_by_monic_hom hg f).coeff ↑i | rfl | theorem | adjoin_root.power_basis_aux'_repr_apply_to_fun | 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"
] | This lemma could be autogenerated by `@[simps]` but unfortunately that would require
unfolding that causes a timeout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_basis' (hg : g.monic) : power_basis R (adjoin_root g) | { gen := root g,
dim := g.nat_degree,
basis := power_basis_aux' hg,
basis_eq_pow := λ i, begin
simp only [power_basis_aux', basis.coe_of_equiv_fun, linear_equiv.coe_symm_mk],
rw finset.sum_eq_single i,
{ rw [function.update_same, monomial_one_right_eq_X_pow, (mk g).map_pow, mk_X] },
{ intros j _ h... | def | adjoin_root.power_basis' | 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",
"basis",
"basis.coe_of_equiv_fun",
"finset.mem_univ",
"linear_equiv.coe_symm_mk",
"map_pow",
"power_basis"
] | The power basis `1, root g, ..., root g ^ (d - 1)` for `adjoin_root g`,
where `g` is a monic polynomial of degree `d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral_root (hf : f ≠ 0) : is_integral K (root f) | is_algebraic_iff_is_integral.mp (is_algebraic_root hf) | lemma | adjoin_root.is_integral_root | 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"
] | [
"is_integral"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly_root (hf : f ≠ 0) : minpoly K (root f) = f * C (f.leading_coeff⁻¹) | begin
have f'_monic : monic _ := monic_mul_leading_coeff_inv hf,
refine (minpoly.unique K _ f'_monic _ _).symm,
{ rw [alg_hom.map_mul, aeval_eq, mk_self, zero_mul] },
intros q q_monic q_aeval,
have commutes : (lift (algebra_map K (adjoin_root f)) (root f) q_aeval).comp (mk q) = mk f,
{ ext,
{ simp only ... | lemma | adjoin_root.minpoly_root | 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",
"alg_hom.map_mul",
"algebra_map",
"inv_eq_zero",
"lift",
"minpoly",
"minpoly.unique",
"ring_hom.comp_apply",
"ring_hom.map_zero",
"with_bot.coe_le_coe",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_basis_aux (hf : f ≠ 0) : basis (fin f.nat_degree) K (adjoin_root f) | begin
set f' := f * C (f.leading_coeff⁻¹) with f'_def,
have deg_f' : f'.nat_degree = f.nat_degree,
{ rw [nat_degree_mul hf, nat_degree_C, add_zero],
{ rwa [ne.def, C_eq_zero, inv_eq_zero, leading_coeff_eq_zero] } },
have minpoly_eq : minpoly K (root f) = f' := minpoly_root hf,
apply @basis.mk _ _ _ (λ (i ... | def | adjoin_root.power_basis_aux | 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",
"basis",
"basis.mk",
"inv_eq_zero",
"linear_independent_pow",
"minpoly"
] | The elements `1, root f, ..., root f ^ (d - 1)` form a basis for `adjoin_root f`,
where `f` is an irreducible polynomial over a field of degree `d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
power_basis (hf : f ≠ 0) :
power_basis K (adjoin_root f) | { gen := root f,
dim := f.nat_degree,
basis := power_basis_aux hf,
basis_eq_pow := basis.mk_apply _ _ } | def | adjoin_root.power_basis | 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",
"basis",
"basis.mk_apply",
"power_basis"
] | The power basis `1, root f, ..., root f ^ (d - 1)` for `adjoin_root f`,
where `f` is an irreducible polynomial over a field of degree `d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minpoly_power_basis_gen (hf : f ≠ 0) :
minpoly K (power_basis hf).gen = f * C (f.leading_coeff⁻¹) | by rw [power_basis_gen, minpoly_root hf] | lemma | adjoin_root.minpoly_power_basis_gen | 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"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly_power_basis_gen_of_monic (hf : f.monic) (hf' : f ≠ 0 := hf.ne_zero) :
minpoly K (power_basis hf').gen = f | by rw [minpoly_power_basis_gen hf', hf.leading_coeff, inv_one, C.map_one, mul_one] | lemma | adjoin_root.minpoly_power_basis_gen_of_monic | 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"
] | [
"inv_one",
"minpoly",
"mul_one",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly.to_adjoin : adjoin_root (minpoly R x) →ₐ[R] adjoin R ({x} : set S) | lift_hom _ ⟨x, self_mem_adjoin_singleton R x⟩
(by simp [← subalgebra.coe_eq_zero, aeval_subalgebra_coe]) | def | adjoin_root.minpoly.to_adjoin | 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",
"minpoly",
"subalgebra.coe_eq_zero"
] | The surjective algebra morphism `R[X]/(minpoly R x) → R[x]`.
If `R` is a GCD domain and `x` is integral, this is an isomorphism,
see `adjoin_root.minpoly.equiv_adjoin`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minpoly.to_adjoin_apply' (a : adjoin_root (minpoly R x)) : minpoly.to_adjoin R x a =
lift_hom (minpoly R x) (⟨x, self_mem_adjoin_singleton R x⟩ : adjoin R ({x} : set S))
(by simp [← subalgebra.coe_eq_zero, aeval_subalgebra_coe]) a | rfl | lemma | adjoin_root.minpoly.to_adjoin_apply' | 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",
"minpoly",
"subalgebra.coe_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly.to_adjoin.apply_X : minpoly.to_adjoin R x (mk (minpoly R x) X) =
⟨x, self_mem_adjoin_singleton R x⟩ | by simp | lemma | adjoin_root.minpoly.to_adjoin.apply_X | 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"
] | [
"minpoly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
minpoly.to_adjoin.surjective : function.surjective (minpoly.to_adjoin R x) | begin
rw [← range_top_iff_surjective, _root_.eq_top_iff, ← adjoin_adjoin_coe_preimage],
refine adjoin_le _,
simp only [alg_hom.coe_range, set.mem_range],
rintro ⟨y₁, y₂⟩ h,
refine ⟨mk (minpoly R x) X, by simpa using h.symm⟩
end | lemma | adjoin_root.minpoly.to_adjoin.surjective | 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"
] | [
"alg_hom.coe_range",
"minpoly",
"set.mem_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv' (h₁ : aeval (root g) (minpoly R pb.gen) = 0) (h₂ : aeval pb.gen g = 0) :
adjoin_root g ≃ₐ[R] S | { to_fun := adjoin_root.lift_hom g pb.gen h₂,
inv_fun := pb.lift (root g) h₁,
left_inv := λ x, induction_on g x $ λ f, by rw [lift_hom_mk, pb.lift_aeval, aeval_eq],
right_inv := λ x, begin
nontriviality S,
obtain ⟨f, hf, rfl⟩ := pb.exists_eq_aeval x,
rw [pb.lift_aeval, aeval_eq, lift_hom_mk]
end,
... | def | adjoin_root.equiv' | 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.lift_hom",
"inv_fun",
"minpoly"
] | If `S` is an extension of `R` with power basis `pb` and `g` is a monic polynomial over `R`
such that `pb.gen` has a minimal polynomial `g`, then `S` is isomorphic to `adjoin_root g`.
Compare `power_basis.equiv_of_root`, which would require
`h₂ : aeval pb.gen (minpoly R (root g)) = 0`; that minimal polynomial is not
gu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv'_to_alg_hom
(h₁ : aeval (root g) (minpoly R pb.gen) = 0) (h₂ : aeval pb.gen g = 0) :
(equiv' g pb h₁ h₂).to_alg_hom = adjoin_root.lift_hom g pb.gen h₂ | rfl | lemma | adjoin_root.equiv'_to_alg_hom | 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.lift_hom",
"minpoly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv'_symm_to_alg_hom
(h₁ : aeval (root g) (minpoly R pb.gen) = 0) (h₂ : aeval pb.gen g = 0) :
(equiv' g pb h₁ h₂).symm.to_alg_hom = pb.lift (root g) h₁ | rfl | lemma | adjoin_root.equiv'_symm_to_alg_hom | 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"
] | [
"minpoly"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv (f : F[X]) (hf : f ≠ 0) :
(adjoin_root f →ₐ[F] L) ≃ {x // x ∈ (f.map (algebra_map F L)).roots} | (power_basis hf).lift_equiv'.trans ((equiv.refl _).subtype_equiv (λ x,
begin
rw [power_basis_gen, minpoly_root hf, polynomial.map_mul, roots_mul,
polynomial.map_C, roots_C, add_zero, equiv.refl_apply],
rw ← polynomial.map_mul, exact map_monic_ne_zero (monic_mul_leading_coeff_inv hf)
end)) | def | adjoin_root.equiv | 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",
"algebra_map",
"equiv",
"equiv.refl",
"equiv.refl_apply",
"polynomial.map_C",
"polynomial.map_mul",
"power_basis"
] | If `L` is a field extension of `F` and `f` is a polynomial over `F` then the set
of maps from `F[x]/(f)` into `L` is in bijection with the set of roots of `f` in `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_map_of_equiv_quot_map_C_map_span_mk :
adjoin_root f ⧸ I.map (of f) ≃+*
adjoin_root f ⧸ (I.map (C : R →+* R[X])).map (span {f})^.quotient.mk | ideal.quot_equiv_of_eq (by rw [of, adjoin_root.mk, ideal.map_map]) | def | adjoin_root.quot_map_of_equiv_quot_map_C_map_span_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",
"adjoin_root.mk",
"ideal.map_map",
"ideal.quot_equiv_of_eq"
] | The natural isomorphism `R[α]/(I[α]) ≅ R[α]/((I[x] ⊔ (f)) / (f))` for `α` a root of
`f : R[X]` and `I : ideal R`.
See `adjoin_root.quot_map_of_equiv` for the isomorphism with `(R/I)[X] / (f mod I)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_map_of_equiv_quot_map_C_map_span_mk_mk (x : adjoin_root f) :
quot_map_of_equiv_quot_map_C_map_span_mk I f (ideal.quotient.mk (I.map (of f)) x) =
ideal.quotient.mk _ x | rfl | lemma | adjoin_root.quot_map_of_equiv_quot_map_C_map_span_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"
] | [
"adjoin_root",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_map_of_equiv_quot_map_C_map_span_mk_symm_mk (x : adjoin_root f) :
(quot_map_of_equiv_quot_map_C_map_span_mk I f).symm
(ideal.quotient.mk ((I.map (C : R →+* R[X])).map (span {f})^.quotient.mk) x) =
ideal.quotient.mk (I.map (of f)) x | by rw [quot_map_of_equiv_quot_map_C_map_span_mk, ideal.quot_equiv_of_eq_symm, quot_equiv_of_eq_mk] | lemma | adjoin_root.quot_map_of_equiv_quot_map_C_map_span_mk_symm_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",
"ideal.quot_equiv_of_eq_symm",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk :
(adjoin_root f) ⧸ (I.map (C : R →+* R[X])).map (span ({f} : set R[X]))^.quotient.mk ≃+*
(R[X] ⧸ I.map (C : R →+* R[X])) ⧸ (span ({f} : set R[X])).map
(I.map (C : R →+* R[X]))^.quotient.mk | quot_quot_equiv_comm (ideal.span ({f} : set R[X])) (I.map (C : R →+* R[X])) | def | adjoin_root.quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_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",
"ideal.span"
] | The natural isomorphism `R[α]/((I[x] ⊔ (f)) / (f)) ≅ (R[x]/I[x])/((f) ⊔ I[x] / I[x])`
for `α` a root of `f : R[X]` and `I : ideal R` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk_mk (p : R[X]) :
quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk I f (ideal.quotient.mk _ (mk f p)) =
quot_quot_mk (I.map C) (span {f}) p | rfl | lemma | adjoin_root.quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_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_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk_symm_quot_quot_mk (p : R[X]) :
(quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk I f).symm
(quot_quot_mk (I.map C) (span {f}) p) = (ideal.quotient.mk _ (mk f p)) | rfl | lemma | adjoin_root.quot_map_C_map_span_mk_equiv_quot_map_C_quot_map_span_mk_symm_quot_quot_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 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.