fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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exist_pow_eq_zero_of_le (p : ℕ) [hchar : ExpChar D p]
{a : D} (ha : a ∉ k) (hinsep : ∀ x : D, IsSeparable k x → x ∈ k):
∃ m, 1 ≤ m ∧ ∀ n, p ^ m ≤ n → (ad k D a)^[n] = 0 := by
obtain ⟨m, hm⟩ := exists_pow_mem_center_of_inseparable' p ha hinsep
refine ⟨m, ⟨hm.1, fun n hn ↦ ?_⟩⟩
have inter : (ad k D a)^[p ^ m] = 0 := by
ext x
rw [ad_eq_lmul_left_sub_lmul_right, ← Module.End.pow_apply, Pi.sub_apply,
sub_pow_expChar_pow_of_commute p m (commute_mulLeft_right a a), sub_apply,
pow_mulLeft, mulLeft_apply, pow_mulRight, mulRight_apply, Pi.zero_apply,
Subring.mem_center_iff.1 hm.2 x]
exact sub_eq_zero_of_eq rfl
rw [(Nat.sub_eq_iff_eq_add hn).1 rfl, Function.iterate_add, inter, Pi.comp_zero,
iterate_map_zero, Function.const_zero]
variable (D) in | lemma | FieldTheory | [
"Mathlib.Algebra.Central.Defs",
"Mathlib.Algebra.CharP.LinearMaps",
"Mathlib.Algebra.CharP.Subring",
"Mathlib.Algebra.GroupWithZero.Conj",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.FieldTheory.PurelyInseparable.Basic"
] | Mathlib/FieldTheory/JacobsonNoether.lean | exist_pow_eq_zero_of_le | If `D` is a purely inseparable extension of `k` of characteristic `p`,
then for every element `a` of `D \ k`, there exists a natural number `m`
greater than 0 such that `(a * x - x * a) ^ n = 0` (as linear maps) for
every `n` greater than `(p ^ m)`. |
exists_separable_and_not_isCentral (H : k ≠ (⊤ : Subring D)) :
∃ x : D, x ∉ k ∧ IsSeparable k x := by
obtain ⟨p, hp⟩ := ExpChar.exists D
by_contra! insep
replace insep : ∀ x : D, IsSeparable k x → x ∈ k :=
fun x h ↦ Classical.byContradiction fun hx ↦ insep x hx h
obtain ⟨a, ha⟩ := not_forall.mp <| mt (Subring.eq_top_iff' k).mpr H
have ha₀ : a ≠ 0 := fun nh ↦ nh ▸ ha <| Subring.zero_mem k
obtain ⟨b, hb1⟩ : ∃ b : D, ad k D a b ≠ 0 := by
rw [Subring.mem_center_iff, not_forall] at ha
use ha.choose
change a * ha.choose - ha.choose * a ≠ 0
simpa only [ne_eq, sub_eq_zero] using Ne.symm ha.choose_spec
obtain ⟨n, hn, hb⟩ : ∃ n, 0 < n ∧ (ad k D a)^[n] b ≠ 0 ∧ (ad k D a)^[n+1] b = 0 := by
obtain ⟨m, -, hm2⟩ := exist_pow_eq_zero_of_le p ha insep
have h_exist : ∃ n, 0 < n ∧ (ad k D a)^[n+1] b = 0 := ⟨p ^ m,
⟨expChar_pow_pos D p m, by rw [hm2 (p ^ m + 1) (Nat.le_add_right _ _), Pi.zero_apply]⟩⟩
classical
refine ⟨Nat.find h_exist, ⟨(Nat.find_spec h_exist).1, ?_, (Nat.find_spec h_exist).2⟩⟩
set t := (Nat.find h_exist - 1 : ℕ) with ht
by_cases h_pos : 0 < t
· convert (ne_eq _ _) ▸ not_and.mp (Nat.find_min h_exist (m := t) (by cutsat)) h_pos
cutsat
· suffices h_find: Nat.find h_exist = 1 by
rwa [h_find]
rw [not_lt, Nat.le_zero, ht, Nat.sub_eq_zero_iff_le] at h_pos
linarith [(Nat.find_spec h_exist).1]
set c := (ad k D a)^[n] b with hc_def
let _ : Invertible c := ⟨c⁻¹, inv_mul_cancel₀ hb.1, mul_inv_cancel₀ hb.1⟩
have hc : a * c = c * a := by
apply eq_of_sub_eq_zero
rw [← mulLeft_apply (R := k), ← mulRight_apply (R := k)]
suffices ad k D a c = 0 from by
rw [← this]; simp [LieRing.of_associative_ring_bracket]
rw [← Function.iterate_succ_apply' (ad k D a) n b, hb.2]
set d := c⁻¹ * a * (ad k D a)^[n - 1] b with hd_def
have hc': c⁻¹ * a = a * c⁻¹ := by
apply_fun (c⁻¹ * · * c⁻¹) at hc
rw [mul_assoc, mul_assoc, mul_inv_cancel₀ hb.1, mul_one, ← mul_assoc,
inv_mul_cancel₀ hb.1, one_mul] at hc
exact hc
have c_eq : a * (ad k D a)^[n - 1] b - (ad k D a)^[n - 1] b * a = c := by
rw [hc_def, ← Nat.sub_add_cancel hn, Function.iterate_succ_apply' (ad k D a) _ b]; rfl
have eq1 : c⁻¹ * a * (ad k D a)^[n - 1] b - c⁻¹ * (ad k D a)^[n - 1] b * a = 1 := by
simp_rw [mul_assoc, (mul_sub_left_distrib c⁻¹ _ _).symm, c_eq, inv_mul_cancel_of_invertible]
have deq : a * d - d * a = a := by
nth_rw 3 [← mul_one a]
rw [hd_def, ← eq1, mul_sub, mul_assoc _ _ a, sub_right_inj, hc',
← mul_assoc, ← mul_assoc, ← mul_assoc]
apply_fun (a⁻¹ * · ) at deq
rw [mul_sub, ← mul_assoc, inv_mul_cancel₀ ha₀, one_mul, ← mul_assoc, sub_eq_iff_eq_add] at deq
... | theorem | FieldTheory | [
"Mathlib.Algebra.Central.Defs",
"Mathlib.Algebra.CharP.LinearMaps",
"Mathlib.Algebra.CharP.Subring",
"Mathlib.Algebra.GroupWithZero.Conj",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.FieldTheory.PurelyInseparable.Basic"
] | Mathlib/FieldTheory/JacobsonNoether.lean | exists_separable_and_not_isCentral | Jacobson-Noether theorem: For a non-commutative division algebra
`D` that is algebraic over its center `k`, there exists an element
`x` of `D \ k` that is separable over `k`. |
exists_separable_and_not_isCentral' {L D : Type*} [Field L] [DivisionRing D]
[Algebra L D] [Algebra.IsAlgebraic L D] [Algebra.IsCentral L D]
(hneq : (⊥ : Subalgebra L D) ≠ ⊤) :
∃ x : D, x ∉ (⊥ : Subalgebra L D) ∧ IsSeparable L x := by
have hcenter : Subalgebra.center L D = ⊥ := le_bot_iff.mp IsCentral.out
have ntrivial : Subring.center D ≠ ⊤ :=
congr(Subalgebra.toSubring $hcenter).trans_ne (Subalgebra.toSubring_injective.ne hneq)
set φ := Subalgebra.equivOfEq (⊥ : Subalgebra L D) (.center L D) hcenter.symm
set equiv : L ≃+* (center D) := ((botEquiv L D).symm.trans φ).toRingEquiv
let _ : Algebra L (center D) := equiv.toRingHom.toAlgebra
let _ : Algebra (center D) L := equiv.symm.toRingHom.toAlgebra
have _ : IsScalarTower L (center D) D := .of_algebraMap_eq fun _ ↦ rfl
have _ : IsScalarTower (center D) L D := .of_algebraMap_eq fun x ↦ by
rw [IsScalarTower.algebraMap_apply L (center D)]
congr
exact (equiv.apply_symm_apply x).symm
have _ : Algebra.IsAlgebraic (center D) D := .tower_top (K := L) _
obtain ⟨x, hxd, hx⟩ := exists_separable_and_not_isCentral D ntrivial
exact ⟨x, ⟨by rwa [← Subalgebra.center_toSubring L, hcenter] at hxd, IsSeparable.tower_top _ hx⟩⟩ | theorem | FieldTheory | [
"Mathlib.Algebra.Central.Defs",
"Mathlib.Algebra.CharP.LinearMaps",
"Mathlib.Algebra.CharP.Subring",
"Mathlib.Algebra.GroupWithZero.Conj",
"Mathlib.Algebra.Lie.OfAssociative",
"Mathlib.FieldTheory.PurelyInseparable.Basic"
] | Mathlib/FieldTheory/JacobsonNoether.lean | exists_separable_and_not_isCentral' | Jacobson-Noether theorem: For a non-commutative division algebra `D`
that is algebraic over a field `L`, if the center of
`D` coincides with `L`, then there exist an element `x` of `D \ L`
that is separable over `L`. |
finiteExts (K : Type*) [Field K] (L : Type*) [Field L] [Algebra K L] :
Set (IntermediateField K L) :=
{E | FiniteDimensional K E} | def | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | finiteExts | Given a field extension `L/K`, `finiteExts K L` is the set of
intermediate field extensions `L/E/K` such that `E/K` is finite. |
fixedByFinite (K L : Type*) [Field K] [Field L] [Algebra K L] : Set (Subgroup (L ≃ₐ[K] L)) :=
IntermediateField.fixingSubgroup '' finiteExts K L
@[deprecated (since := "2025-03-16")]
alias IntermediateField.finiteDimensional_bot := IntermediateField.instFiniteSubtypeMemBot
@[deprecated (since := "2025-03-12")]
alias IntermediateField.fixingSubgroup.bot := IntermediateField.fixingSubgroup_bot | def | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | fixedByFinite | Given a field extension `L/K`, `fixedByFinite K L` is the set of
subsets `Gal(L/E)` of `L ≃ₐ[K] L`, where `E/K` is finite. |
top_fixedByFinite {K L : Type*} [Field K] [Field L] [Algebra K L] :
⊤ ∈ fixedByFinite K L :=
⟨⊥, IntermediateField.instFiniteSubtypeMemBot K, IntermediateField.fixingSubgroup_bot⟩
@[deprecated (since := "2025-03-16")]
alias finiteDimensional_sup := IntermediateField.finiteDimensional_sup | theorem | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | top_fixedByFinite | If `L/K` is a field extension, then we have `Gal(L/K) ∈ fixedByFinite K L`. |
galBasis (K L : Type*) [Field K] [Field L] [Algebra K L] : FilterBasis (L ≃ₐ[K] L) where
sets := (fun g => g.carrier) '' fixedByFinite K L
nonempty := ⟨⊤, ⊤, top_fixedByFinite, rfl⟩
inter_sets := by
rintro _ _ ⟨_, ⟨E1, h_E1, rfl⟩, rfl⟩ ⟨_, ⟨E2, h_E2, rfl⟩, rfl⟩
have : FiniteDimensional K E1 := h_E1
have : FiniteDimensional K E2 := h_E2
refine ⟨(E1 ⊔ E2).fixingSubgroup.carrier, ⟨_, ⟨_, E1.finiteDimensional_sup E2, rfl⟩, rfl⟩, ?_⟩
exact Set.subset_inter (E1.fixingSubgroup_le le_sup_left) (E2.fixingSubgroup_le le_sup_right) | def | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | galBasis | Given a field extension `L/K`, `galBasis K L` is the filter basis on `L ≃ₐ[K] L` whose sets
are `Gal(L/E)` for intermediate fields `E` with `E/K` finite dimensional. |
mem_galBasis_iff (K L : Type*) [Field K] [Field L] [Algebra K L] (U : Set (L ≃ₐ[K] L)) :
U ∈ galBasis K L ↔ U ∈ (fun g => g.carrier) '' fixedByFinite K L :=
Iff.rfl | theorem | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | mem_galBasis_iff | A subset of `L ≃ₐ[K] L` is a member of `galBasis K L` if and only if it is the underlying set
of `Gal(L/E)` for some finite subextension `E/K`. |
galGroupBasis (K L : Type*) [Field K] [Field L] [Algebra K L] :
GroupFilterBasis (L ≃ₐ[K] L) where
toFilterBasis := galBasis K L
one' := fun ⟨H, _, h2⟩ => h2 ▸ H.one_mem
mul' {U} hU :=
⟨U, hU, by
rcases hU with ⟨H, _, rfl⟩
rintro x ⟨a, haH, b, hbH, rfl⟩
exact H.mul_mem haH hbH⟩
inv' {U} hU :=
⟨U, hU, by
rcases hU with ⟨H, _, rfl⟩
exact fun _ => H.inv_mem'⟩
conj' := by
rintro σ U ⟨H, ⟨E, hE, rfl⟩, rfl⟩
let F : IntermediateField K L := E.map σ.symm.toAlgHom
refine ⟨F.fixingSubgroup.carrier, ⟨⟨F.fixingSubgroup, ⟨F, ?_, rfl⟩, rfl⟩, fun g hg => ?_⟩⟩
· have : FiniteDimensional K E := hE
exact IntermediateField.finiteDimensional_map σ.symm.toAlgHom
change σ * g * σ⁻¹ ∈ E.fixingSubgroup
rw [IntermediateField.mem_fixingSubgroup_iff]
intro x hx
change σ (g (σ⁻¹ x)) = x
have h_in_F : σ⁻¹ x ∈ F := ⟨x, hx, by dsimp; rw [← AlgEquiv.invFun_eq_symm]; rfl⟩
have h_g_fix : g (σ⁻¹ x) = σ⁻¹ x := by
rw [Subgroup.mem_carrier, IntermediateField.mem_fixingSubgroup_iff F g] at hg
exact hg (σ⁻¹ x) h_in_F
rw [h_g_fix]
change σ (σ⁻¹ x) = x
exact AlgEquiv.apply_symm_apply σ x | def | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | galGroupBasis | For a field extension `L/K`, `galGroupBasis K L` is the group filter basis on `L ≃ₐ[K] L`
whose sets are `Gal(L/E)` for finite subextensions `E/K`. |
krullTopology (K L : Type*) [Field K] [Field L] [Algebra K L] :
TopologicalSpace (L ≃ₐ[K] L) :=
GroupFilterBasis.topology (galGroupBasis K L) | instance | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | krullTopology | For a field extension `L/K`, `krullTopology K L` is the topological space structure on
`L ≃ₐ[K] L` induced by the group filter basis `galGroupBasis K L`. |
IntermediateField.fixingSubgroup_isOpen {K L : Type*} [Field K] [Field L] [Algebra K L]
(E : IntermediateField K L) [FiniteDimensional K E] :
IsOpen (E.fixingSubgroup : Set (L ≃ₐ[K] L)) := by
have h_basis : E.fixingSubgroup.carrier ∈ galGroupBasis K L :=
⟨E.fixingSubgroup, ⟨E, ‹_›, rfl⟩, rfl⟩
have h_nhds := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
exact Subgroup.isOpen_of_mem_nhds _ h_nhds | theorem | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | IntermediateField.fixingSubgroup_isOpen | For a field extension `L/K`, the Krull topology on `L ≃ₐ[K] L` makes it a topological group. -/
@[stacks 0BMJ "We define Krull topology directly without proving the universal property"]
instance (K L : Type*) [Field K] [Field L] [Algebra K L] : IsTopologicalGroup (L ≃ₐ[K] L) :=
GroupFilterBasis.isTopologicalGroup (galGroupBasis K L)
open scoped Topology in
lemma krullTopology_mem_nhds_one_iff (K L : Type*) [Field K] [Field L] [Algebra K L]
(s : Set (L ≃ₐ[K] L)) : s ∈ 𝓝 1 ↔ ∃ E : IntermediateField K L,
FiniteDimensional K E ∧ (E.fixingSubgroup : Set (L ≃ₐ[K] L)) ⊆ s := by
rw [GroupFilterBasis.nhds_one_eq]
constructor
· rintro ⟨-, ⟨-, ⟨E, fin, rfl⟩, rfl⟩, hE⟩
exact ⟨E, fin, hE⟩
· rintro ⟨E, fin, hE⟩
exact ⟨E.fixingSubgroup, ⟨E.fixingSubgroup, ⟨E, fin, rfl⟩, rfl⟩, hE⟩
open scoped Topology in
lemma krullTopology_mem_nhds_one_iff_of_normal (K L : Type*) [Field K] [Field L] [Algebra K L]
[Normal K L] (s : Set (L ≃ₐ[K] L)) : s ∈ 𝓝 1 ↔ ∃ E : IntermediateField K L,
FiniteDimensional K E ∧ Normal K E ∧ (E.fixingSubgroup : Set (L ≃ₐ[K] L)) ⊆ s := by
rw [krullTopology_mem_nhds_one_iff]
refine ⟨fun ⟨E, _, hE⟩ ↦ ?_, fun ⟨E, hE⟩ ↦ ⟨E, hE.1, hE.2.2⟩⟩
use (IntermediateField.normalClosure K E L)
simp only [normalClosure.is_finiteDimensional K E L, normalClosure.normal K E L, true_and]
exact le_trans (E.fixingSubgroup_antitone E.le_normalClosure) hE
section KrullT2
open scoped Topology Filter
/-- Let `L/E/K` be a tower of fields with `E/K` finite. Then `Gal(L/E)` is an open subgroup of
`L ≃ₐ[K] L`. |
IntermediateField.fixingSubgroup_isClosed {K L : Type*} [Field K] [Field L] [Algebra K L]
(E : IntermediateField K L) [FiniteDimensional K E] :
IsClosed (E.fixingSubgroup : Set (L ≃ₐ[K] L)) :=
OpenSubgroup.isClosed ⟨E.fixingSubgroup, E.fixingSubgroup_isOpen⟩ | theorem | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | IntermediateField.fixingSubgroup_isClosed | Given a tower of fields `L/E/K`, with `E/K` finite, the subgroup `Gal(L/E) ≤ L ≃ₐ[K] L` is
closed. |
krullTopology_t2 {K L : Type*} [Field K] [Field L] [Algebra K L]
[Algebra.IsIntegral K L] : T2Space (L ≃ₐ[K] L) :=
{ t2 := fun f g hfg => by
let φ := f⁻¹ * g
obtain ⟨x, hx⟩ := DFunLike.exists_ne hfg
have hφx : φ x ≠ x := by
apply ne_of_apply_ne f
change f (f.symm (g x)) ≠ f x
rw [AlgEquiv.apply_symm_apply f (g x), ne_comm]
exact hx
let E : IntermediateField K L := IntermediateField.adjoin K {x}
let h_findim : FiniteDimensional K E := IntermediateField.adjoin.finiteDimensional
(Algebra.IsIntegral.isIntegral x)
let H := E.fixingSubgroup
have h_basis : (H : Set (L ≃ₐ[K] L)) ∈ galGroupBasis K L := ⟨H, ⟨E, ⟨h_findim, rfl⟩⟩, rfl⟩
have h_nhds := GroupFilterBasis.mem_nhds_one (galGroupBasis K L) h_basis
rw [mem_nhds_iff] at h_nhds
rcases h_nhds with ⟨W, hWH, hW_open, hW_1⟩
refine ⟨f • W, g • W,
⟨hW_open.leftCoset f, hW_open.leftCoset g, ⟨1, hW_1, mul_one _⟩, ⟨1, hW_1, mul_one _⟩, ?_⟩⟩
rw [Set.disjoint_left]
rintro σ ⟨w1, hw1, h⟩ ⟨w2, hw2, rfl⟩
dsimp at h
rw [eq_inv_mul_iff_mul_eq.symm, ← mul_assoc, mul_inv_eq_iff_eq_mul.symm] at h
have h_in_H : w1 * w2⁻¹ ∈ H := H.mul_mem (hWH hw1) (H.inv_mem (hWH hw2))
rw [h] at h_in_H
change φ ∈ E.fixingSubgroup at h_in_H
rw [IntermediateField.mem_fixingSubgroup_iff] at h_in_H
specialize h_in_H x
have hxE : x ∈ E := by
apply IntermediateField.subset_adjoin
apply Set.mem_singleton
exact hφx (h_in_H hxE) } | theorem | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | krullTopology_t2 | If `L/K` is an algebraic extension, then the Krull topology on `L ≃ₐ[K] L` is Hausdorff. |
krullTopology_isTotallySeparated {K L : Type*} [Field K] [Field L] [Algebra K L]
[Algebra.IsIntegral K L] : IsTotallySeparated (Set.univ : Set (L ≃ₐ[K] L)) :=
(totallySeparatedSpace_iff _).mp inferInstance
@[deprecated (since := "2025-04-03")]
alias krullTopology_totallyDisconnected := krullTopology_isTotallySeparated | theorem | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | krullTopology_isTotallySeparated | If `L/K` is an algebraic field extension, then the Krull topology on `L ≃ₐ[K] L` is
totally disconnected. |
krullTopology_discreteTopology_of_finiteDimensional (K L : Type*) [Field K] [Field L]
[Algebra K L] [FiniteDimensional K L] : DiscreteTopology (L ≃ₐ[K] L) := by
rw [discreteTopology_iff_isOpen_singleton_one]
change IsOpen ((⊥ : Subgroup (L ≃ₐ[K] L)) : Set (L ≃ₐ[K] L))
rw [← IntermediateField.fixingSubgroup_top]
exact IntermediateField.fixingSubgroup_isOpen ⊤ | instance | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | krullTopology_discreteTopology_of_finiteDimensional | null |
map_fixingSubgroup [Normal k E] :
(L.map (IsScalarTower.toAlgHom k E K)).fixingSubgroup =
L.fixingSubgroup.comap (AlgEquiv.restrictNormalHom (F := k) (K₁ := K) E) := by
ext f
simp only [Subgroup.mem_comap, mem_fixingSubgroup_iff]
constructor
· rintro h x hx
change f.restrictNormal E x = x
apply_fun _ using (algebraMap E K).injective
rw [AlgEquiv.restrictNormal_commutes]
exact h _ ⟨x, hx, rfl⟩
· rintro h _ ⟨x, hx, rfl⟩
replace h := congr(algebraMap E K $(show f.restrictNormal E x = x from h x hx))
rwa [AlgEquiv.restrictNormal_commutes] at h | theorem | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | map_fixingSubgroup | If `K / E / k` is a field extension tower with `E / k` normal,
`L` is an intermediate field of `E / k`, then the fixing subgroup of `L` viewed as an
intermediate field of `K / k` is equal to the preimage of the fixing subgroup of `L` viewed as an
intermediate field of `E / k` under the natural map `Aut(K / k) → Aut(E / k)`
(`AlgEquiv.restrictNormalHom`). |
map_fixingSubgroup_index [Normal k E] [Normal k K] :
(L.map (IsScalarTower.toAlgHom k E K)).fixingSubgroup.index = L.fixingSubgroup.index := by
rw [L.map_fixingSubgroup K, L.fixingSubgroup.index_comap_of_surjective
(AlgEquiv.restrictNormalHom_surjective _)]
variable {K} in | theorem | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | map_fixingSubgroup_index | If `K / E / k` is a field extension tower with `E / k` and `K / k` normal,
`L` is an intermediate field of `E / k`, then the index of the fixing subgroup of `L` viewed as an
intermediate field of `K / k` is equal to the index of the fixing subgroup of `L` viewed as an
intermediate field of `E / k`. |
finrank_eq_fixingSubgroup_index (L : IntermediateField k K) [IsGalois k K] :
Module.finrank k L = L.fixingSubgroup.index := by
wlog hnfd : FiniteDimensional k L generalizing L
· rw [Module.finrank_of_infinite_dimensional hnfd]
by_contra! h
replace h : L.fixingSubgroup.FiniteIndex := ⟨h.symm⟩
obtain ⟨L', hfd, hL'⟩ :=
exists_lt_finrank_of_infinite_dimensional hnfd L.fixingSubgroup.index
let i := (liftAlgEquiv L').toLinearEquiv
replace hfd := i.finiteDimensional
rw [i.finrank_eq, this _ hfd] at hL'
exact (Subgroup.index_antitone <| fixingSubgroup_le <|
IntermediateField.lift_le L').not_gt hL'
let E := normalClosure k L K
have hle : L ≤ E := by simpa only [fieldRange_val] using L.val.fieldRange_le_normalClosure
let L' := restrict hle
have h := Module.finrank_mul_finrank k ↥L' ↥E
classical
rw [← IsGalois.card_fixingSubgroup_eq_finrank L', ← IsGalois.card_aut_eq_finrank k E] at h
rw [← L'.fixingSubgroup.index_mul_card, Nat.mul_left_inj Finite.card_pos.ne'] at h
rw [(restrict_algEquiv hle).toLinearEquiv.finrank_eq, h, ← L'.map_fixingSubgroup_index K]
congr 2
exact lift_restrict hle | theorem | FieldTheory | [
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.Topology.Algebra.FilterBasis",
"Mathlib.Topology.Algebra.OpenSubgroup"
] | Mathlib/FieldTheory/KrullTopology.lean | finrank_eq_fixingSubgroup_index | If `K / k` is a Galois extension, `L` is an intermediate field of `K / k`, then `[L : k]`
as a natural number is equal to the index of the fixing subgroup of `L`. |
X_pow_sub_C_splits_of_isPrimitiveRoot
{n : ℕ} {ζ : K} (hζ : IsPrimitiveRoot ζ n) {α a : K} (e : α ^ n = a) :
(X ^ n - C a).Splits (RingHom.id _) := by
cases n.eq_zero_or_pos with
| inl hn =>
rw [hn, pow_zero, ← C.map_one, ← map_sub]
exact splits_C _ _
| inr hn =>
rw [splits_iff_card_roots, ← nthRoots, hζ.card_nthRoots, natDegree_X_pow_sub_C, if_pos ⟨α, e⟩] | theorem | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | X_pow_sub_C_splits_of_isPrimitiveRoot | null |
private
X_pow_sub_C_eq_prod'
{n : ℕ} {ζ : K} (hζ : IsPrimitiveRoot ζ n) {α a : K} (hn : 0 < n) (e : α ^ n = a) :
(X ^ n - C a) = ∏ i ∈ Finset.range n, (X - C (ζ ^ i * α)) := by
rw [eq_prod_roots_of_monic_of_splits_id (monic_X_pow_sub_C _ (Nat.pos_iff_ne_zero.mp hn))
(X_pow_sub_C_splits_of_isPrimitiveRoot hζ e), ← nthRoots, hζ.nthRoots_eq e, Multiset.map_map]
rfl | theorem | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | X_pow_sub_C_eq_prod' | null |
X_pow_sub_C_eq_prod {R : Type*} [CommRing R] [IsDomain R]
{n : ℕ} {ζ : R} (hζ : IsPrimitiveRoot ζ n) {α a : R} (hn : 0 < n) (e : α ^ n = a) :
(X ^ n - C a) = ∏ i ∈ Finset.range n, (X - C (ζ ^ i * α)) := by
let K := FractionRing R
let i := algebraMap R K
have h := FaithfulSMul.algebraMap_injective R K
apply_fun Polynomial.map i using map_injective i h
simpa only [Polynomial.map_sub, Polynomial.map_pow, map_X, map_C, map_mul, map_pow,
Polynomial.map_prod, Polynomial.map_mul]
using X_pow_sub_C_eq_prod' (hζ.map_of_injective h) hn <| map_pow i α n ▸ congrArg i e | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | X_pow_sub_C_eq_prod | null |
X_pow_mul_sub_C_irreducible
{n m : ℕ} {a : K} (hm : Irreducible (X ^ m - C a))
(hn : ∀ (E : Type u) [Field E] [Algebra K E] (x : E) (_ : minpoly K x = X ^ m - C a),
Irreducible (X ^ n - C (AdjoinSimple.gen K x))) :
Irreducible (X ^ (n * m) - C a) := by
have hm' : m ≠ 0 := by
rintro rfl
rw [pow_zero, ← C.map_one, ← map_sub] at hm
exact not_irreducible_C _ hm
simpa [pow_mul] using irreducible_comp (monic_X_pow_sub_C a hm') (monic_X_pow n) hm
(by simpa only [Polynomial.map_pow, map_X] using hn) | theorem | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | X_pow_mul_sub_C_irreducible | null |
X_pow_sub_C_irreducible_of_odd
{n : ℕ} (hn : Odd n) {a : K} (ha : ∀ p : ℕ, p.Prime → p ∣ n → ∀ b : K, b ^ p ≠ a) :
Irreducible (X ^ n - C a) := by
induction n using induction_on_primes generalizing K a with
| zero => simp [← Nat.not_even_iff_odd] at hn
| one => simpa using irreducible_X_sub_C a
| prime_mul p n hp IH =>
rw [mul_comm]
apply X_pow_mul_sub_C_irreducible
(X_pow_sub_C_irreducible_of_prime hp (ha p hp (dvd_mul_right _ _)))
intro E _ _ x hx
have : IsIntegral K x := not_not.mp fun h ↦ by
simpa only [degree_zero, degree_X_pow_sub_C hp.pos,
WithBot.natCast_ne_bot] using congr_arg degree (hx.symm.trans (dif_neg h))
apply IH (Nat.odd_mul.mp hn).2
intro q hq hqn b hb
apply ha q hq (dvd_mul_of_dvd_right hqn p) (Algebra.norm _ b)
rw [← map_pow, hb, ← adjoin.powerBasis_gen this,
Algebra.PowerBasis.norm_gen_eq_coeff_zero_minpoly]
simp [minpoly_gen, hx, hp.ne_zero.symm, (Nat.odd_mul.mp hn).1.neg_pow] | theorem | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | X_pow_sub_C_irreducible_of_odd | null |
X_pow_sub_C_irreducible_iff_forall_prime_of_odd {n : ℕ} (hn : Odd n) {a : K} :
Irreducible (X ^ n - C a) ↔ (∀ p : ℕ, p.Prime → p ∣ n → ∀ b : K, b ^ p ≠ a) :=
⟨fun e _ hp hpn ↦ pow_ne_of_irreducible_X_pow_sub_C e hpn hp.ne_one,
X_pow_sub_C_irreducible_of_odd hn⟩ | theorem | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | X_pow_sub_C_irreducible_iff_forall_prime_of_odd | null |
X_pow_sub_C_irreducible_iff_of_odd {n : ℕ} (hn : Odd n) {a : K} :
Irreducible (X ^ n - C a) ↔ (∀ d, d ∣ n → d ≠ 1 → ∀ b : K, b ^ d ≠ a) :=
⟨fun e _ ↦ pow_ne_of_irreducible_X_pow_sub_C e,
fun H ↦ X_pow_sub_C_irreducible_of_odd hn fun p hp hpn ↦ (H p hpn hp.ne_one)⟩ | theorem | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | X_pow_sub_C_irreducible_iff_of_odd | null |
X_pow_sub_C_irreducible_of_prime_pow
{p : ℕ} (hp : p.Prime) (hp' : p ≠ 2) (n : ℕ) {a : K} (ha : ∀ b : K, b ^ p ≠ a) :
Irreducible (X ^ (p ^ n) - C a) := by
apply X_pow_sub_C_irreducible_of_odd (hp.odd_of_ne_two hp').pow
intro q hq hq'
simpa [(Nat.prime_dvd_prime_iff_eq hq hp).mp (hq.dvd_of_dvd_pow hq')] using ha | theorem | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | X_pow_sub_C_irreducible_of_prime_pow | null |
X_pow_sub_C_irreducible_iff_of_prime_pow
{p : ℕ} (hp : p.Prime) (hp' : p ≠ 2) {n} (hn : n ≠ 0) {a : K} :
Irreducible (X ^ p ^ n - C a) ↔ ∀ b, b ^ p ≠ a :=
⟨(pow_ne_of_irreducible_X_pow_sub_C · (dvd_pow dvd_rfl hn) hp.ne_one),
X_pow_sub_C_irreducible_of_prime_pow hp hp' n⟩ | theorem | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | X_pow_sub_C_irreducible_iff_of_prime_pow | null |
Polynomial.separable_X_pow_sub_C_of_irreducible : (X ^ n - C a).Separable := by
letI := Fact.mk H
letI : Algebra K K[n√a] := inferInstance
have hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)
by_cases hn' : n = 1
· rw [hn', pow_one]; exact separable_X_sub_C
have ⟨ζ, hζ⟩ := hζ
rw [mem_primitiveRoots (Nat.pos_of_ne_zero <| ne_zero_of_irreducible_X_pow_sub_C H)] at hζ
rw [← separable_map (algebraMap K K[n√a]), Polynomial.map_sub, Polynomial.map_pow, map_C, map_X,
AdjoinRoot.algebraMap_eq,
X_pow_sub_C_eq_prod (hζ.map_of_injective (algebraMap K _).injective) hn
(root_X_pow_sub_C_pow n a), separable_prod_X_sub_C_iff']
exact (hζ.map_of_injective (algebraMap K K[n√a]).injective).injOn_pow_mul
(root_X_pow_sub_C_ne_zero (lt_of_le_of_ne (show 1 ≤ n from hn) (Ne.symm hn')) _)
variable (n) | theorem | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | Polynomial.separable_X_pow_sub_C_of_irreducible | Also see `Polynomial.separable_X_pow_sub_C_unit` |
noncomputable
autAdjoinRootXPowSubCHom :
rootsOfUnity n K →* (K[n√a] →ₐ[K] K[n√a]) where
toFun := fun η ↦ liftHom (X ^ n - C a) (((η : Kˣ) : K) • (root _) : K[n√a]) <| by
have := (mem_rootsOfUnity' _ _).mp η.prop
rw [map_sub, map_pow, aeval_C, aeval_X, Algebra.smul_def, mul_pow, root_X_pow_sub_C_pow,
AdjoinRoot.algebraMap_eq, ← map_pow, this, map_one, one_mul, sub_self]
map_one' := algHom_ext <| by simp
map_mul' := fun ε η ↦ algHom_ext <| by simp [mul_smul, smul_comm ((ε : Kˣ) : K)] | def | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autAdjoinRootXPowSubCHom | The natural embedding of the roots of unity of `K` into `Gal(K[ⁿ√a]/K)`, by sending
`η ↦ (ⁿ√a ↦ η • ⁿ√a)`. Also see `autAdjoinRootXPowSubC` for the `AlgEquiv` version. |
noncomputable
autAdjoinRootXPowSubC :
rootsOfUnity n K →* (K[n√a] ≃ₐ[K] K[n√a]) :=
(AlgEquiv.algHomUnitsEquiv _ _).toMonoidHom.comp (autAdjoinRootXPowSubCHom n a).toHomUnits
variable {n} | def | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autAdjoinRootXPowSubC | The natural embedding of the roots of unity of `K` into `Gal(K[ⁿ√a]/K)`, by sending
`η ↦ (ⁿ√a ↦ η • ⁿ√a)`. This is an isomorphism when `K` contains a primitive root of unity.
See `autAdjoinRootXPowSubCEquiv`. |
autAdjoinRootXPowSubC_root (η) :
autAdjoinRootXPowSubC n a η (root _) = ((η : Kˣ) : K) • root _ := by
dsimp [autAdjoinRootXPowSubC, autAdjoinRootXPowSubCHom, AlgEquiv.algHomUnitsEquiv]
apply liftHom_root
variable {a} | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autAdjoinRootXPowSubC_root | null |
noncomputable
AdjoinRootXPowSubCEquivToRootsOfUnity [NeZero n] (σ : K[n√a] ≃ₐ[K] K[n√a]) :
rootsOfUnity n K :=
letI := Fact.mk H
letI : IsDomain K[n√a] := inferInstance
letI := Classical.decEq K
(rootsOfUnityEquivOfPrimitiveRoots (n := n) (algebraMap K K[n√a]).injective hζ).symm
(rootsOfUnity.mkOfPowEq (if a = 0 then 1 else σ (root _) / root _) (by
split
· exact one_pow _
rw [div_pow, ← map_pow]
simp only [root_X_pow_sub_C_pow, ← AdjoinRoot.algebraMap_eq, AlgEquiv.commutes]
rw [div_self]
rwa [Ne, map_eq_zero_iff _ (algebraMap K _).injective])) | def | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | AdjoinRootXPowSubCEquivToRootsOfUnity | The inverse function of `autAdjoinRootXPowSubC` if `K` has all roots of unity.
See `autAdjoinRootXPowSubCEquiv`. |
noncomputable
autAdjoinRootXPowSubCEquiv [NeZero n] :
rootsOfUnity n K ≃* (K[n√a] ≃ₐ[K] K[n√a]) where
__ := autAdjoinRootXPowSubC n a
invFun := AdjoinRootXPowSubCEquivToRootsOfUnity hζ H
left_inv := by
intro η
have := Fact.mk H
have : IsDomain K[n√a] := inferInstance
letI : Algebra K K[n√a] := inferInstance
apply (rootsOfUnityEquivOfPrimitiveRoots (algebraMap K K[n√a]).injective hζ).injective
ext
simp only [AdjoinRoot.algebraMap_eq, OneHom.toFun_eq_coe, MonoidHom.toOneHom_coe,
autAdjoinRootXPowSubC_root, Algebra.smul_def, MulEquiv.apply_symm_apply,
rootsOfUnity.val_mkOfPowEq_coe, val_rootsOfUnityEquivOfPrimitiveRoots_apply_coe,
AdjoinRootXPowSubCEquivToRootsOfUnity]
split_ifs with h
· obtain rfl := not_imp_not.mp (fun hn ↦ ne_zero_of_irreducible_X_pow_sub_C' hn H) h
have : (η : Kˣ) = 1 := (pow_one _).symm.trans η.prop
simp only [this, Units.val_one, map_one]
· exact mul_div_cancel_right₀ _ (root_X_pow_sub_C_ne_zero' (NeZero.pos n) h)
right_inv := by
intro e
have := Fact.mk H
letI : Algebra K K[n√a] := inferInstance
apply AlgEquiv.coe_algHom_injective
apply AdjoinRoot.algHom_ext
simp only [AdjoinRootXPowSubCEquivToRootsOfUnity, AdjoinRoot.algebraMap_eq, OneHom.toFun_eq_coe,
MonoidHom.toOneHom_coe, AlgHom.coe_coe, autAdjoinRootXPowSubC_root, Algebra.smul_def]
rw [rootsOfUnityEquivOfPrimitiveRoots_symm_apply, rootsOfUnity.val_mkOfPowEq_coe]
split_ifs with h
· obtain rfl := not_imp_not.mp (fun hn ↦ ne_zero_of_irreducible_X_pow_sub_C' hn H) h
rw [(pow_one _).symm.trans (root_X_pow_sub_C_pow 1 a), one_mul,
← AdjoinRoot.algebraMap_eq, AlgEquiv.commutes]
· refine div_mul_cancel₀ _ (root_X_pow_sub_C_ne_zero' (NeZero.pos n) h) | def | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autAdjoinRootXPowSubCEquiv | The equivalence between the roots of unity of `K` and `Gal(K[ⁿ√a]/K)`. |
autAdjoinRootXPowSubCEquiv_root [NeZero n] (η) :
autAdjoinRootXPowSubCEquiv hζ H η (root _) = ((η : Kˣ) : K) • root _ :=
autAdjoinRootXPowSubC_root a η | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autAdjoinRootXPowSubCEquiv_root | null |
autAdjoinRootXPowSubCEquiv_symm_smul [NeZero n] (σ) :
((autAdjoinRootXPowSubCEquiv hζ H).symm σ : Kˣ) • (root _ : K[n√a]) = σ (root _) := by
have := Fact.mk H
simp only [autAdjoinRootXPowSubCEquiv, OneHom.toFun_eq_coe, MonoidHom.toOneHom_coe,
MulEquiv.symm_mk, MulEquiv.coe_mk, Equiv.coe_fn_symm_mk, AdjoinRootXPowSubCEquivToRootsOfUnity,
AdjoinRoot.algebraMap_eq, rootsOfUnity.mkOfPowEq, Units.smul_def, Algebra.smul_def,
rootsOfUnityEquivOfPrimitiveRoots_symm_apply, Units.val_ofPowEqOne, ite_mul, one_mul]
simp_rw [← root_X_pow_sub_C_eq_zero_iff H]
split_ifs with h
· rw [h, map_zero]
· rw [div_mul_cancel₀ _ h] | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autAdjoinRootXPowSubCEquiv_symm_smul | null |
isSplittingField_AdjoinRoot_X_pow_sub_C :
haveI := Fact.mk H
letI : Algebra K K[n√a] := inferInstance
IsSplittingField K K[n√a] (X ^ n - C a) := by
have := Fact.mk H
letI : Algebra K K[n√a] := inferInstance
constructor
· rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_C,
Polynomial.map_X]
have ⟨_, hζ⟩ := hζ
rw [mem_primitiveRoots (Nat.pos_of_ne_zero <| ne_zero_of_irreducible_X_pow_sub_C H)] at hζ
exact X_pow_sub_C_splits_of_isPrimitiveRoot (hζ.map_of_injective (algebraMap K _).injective)
(root_X_pow_sub_C_pow n a)
· rw [eq_top_iff, ← AdjoinRoot.adjoinRoot_eq_top]
apply Algebra.adjoin_mono
have := ne_zero_of_irreducible_X_pow_sub_C H
rw [Set.singleton_subset_iff, mem_rootSet_of_ne (X_pow_sub_C_ne_zero
(Nat.pos_of_ne_zero this) a), aeval_def, AdjoinRoot.algebraMap_eq, AdjoinRoot.eval₂_root]
variable {α : L} (hα : α ^ n = algebraMap K L a) | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | isSplittingField_AdjoinRoot_X_pow_sub_C | null |
noncomputable
adjoinRootXPowSubCEquiv (hζ : (primitiveRoots n K).Nonempty) (H : Irreducible (X ^ n - C a))
(hα : α ^ n = algebraMap K L a) : K[n√a] ≃ₐ[K] L :=
AlgEquiv.ofBijective (AdjoinRoot.liftHom (X ^ n - C a) α (by simp [hα])) <| by
haveI := Fact.mk H
letI := isSplittingField_AdjoinRoot_X_pow_sub_C hζ H
refine ⟨(liftHom (X ^ n - C a) α _).injective, ?_⟩
rw [← AlgHom.range_eq_top, ← IsSplittingField.adjoin_rootSet _ (X ^ n - C a),
eq_comm, adjoin_rootSet_eq_range, IsSplittingField.adjoin_rootSet]
exact IsSplittingField.splits _ _ | def | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | adjoinRootXPowSubCEquiv | Suppose `L/K` is the splitting field of `Xⁿ - a`, then a choice of `ⁿ√a` gives an equivalence of
`L` with `K[n√a]`. |
adjoinRootXPowSubCEquiv_root :
adjoinRootXPowSubCEquiv hζ H hα (root _) = α := by
rw [adjoinRootXPowSubCEquiv, AlgEquiv.coe_ofBijective, liftHom_root] | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | adjoinRootXPowSubCEquiv_root | null |
adjoinRootXPowSubCEquiv_symm_eq_root :
(adjoinRootXPowSubCEquiv hζ H hα).symm α = root _ := by
apply (adjoinRootXPowSubCEquiv hζ H hα).injective
rw [(adjoinRootXPowSubCEquiv hζ H hα).apply_symm_apply, adjoinRootXPowSubCEquiv_root]
include hζ H hα in | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | adjoinRootXPowSubCEquiv_symm_eq_root | null |
Algebra.adjoin_root_eq_top_of_isSplittingField :
Algebra.adjoin K {α} = ⊤ := by
apply Subalgebra.map_injective (B := K[n√a]) (f := (adjoinRootXPowSubCEquiv hζ H hα).symm)
(adjoinRootXPowSubCEquiv hζ H hα).symm.injective
rw [Algebra.map_top, (AlgHom.range_eq_top _).mpr
(adjoinRootXPowSubCEquiv hζ H hα).symm.surjective, AlgHom.map_adjoin,
Set.image_singleton, AlgHom.coe_coe, adjoinRootXPowSubCEquiv_symm_eq_root, adjoinRoot_eq_top]
include hζ H hα in | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | Algebra.adjoin_root_eq_top_of_isSplittingField | null |
IntermediateField.adjoin_root_eq_top_of_isSplittingField :
K⟮α⟯ = ⊤ := by
refine (IntermediateField.eq_adjoin_of_eq_algebra_adjoin _ _ _ ?_).symm
exact (Algebra.adjoin_root_eq_top_of_isSplittingField hζ H hα).symm
variable (a) (L) | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | IntermediateField.adjoin_root_eq_top_of_isSplittingField | null |
noncomputable
rootOfSplitsXPowSubC (hn : 0 < n) (a : K)
(L) [Field L] [Algebra K L] [IsSplittingField K L (X ^ n - C a)] : L :=
(rootOfSplits _ (IsSplittingField.splits L (X ^ n - C a))
(by simpa [degree_X_pow_sub_C hn] using Nat.pos_iff_ne_zero.mp hn)) | abbrev | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | rootOfSplitsXPowSubC | An arbitrary choice of `ⁿ√a` in the splitting field of `Xⁿ - a`. |
rootOfSplitsXPowSubC_pow [NeZero n] :
(rootOfSplitsXPowSubC (NeZero.pos n) a L) ^ n = algebraMap K L a := by
have := map_rootOfSplits _ (IsSplittingField.splits L (X ^ n - C a))
simp only [eval₂_sub, eval₂_X_pow, eval₂_C, sub_eq_zero] at this
exact this _
variable {a} | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | rootOfSplitsXPowSubC_pow | null |
noncomputable
autEquivRootsOfUnity [NeZero n] :
(L ≃ₐ[K] L) ≃* (rootsOfUnity n K) :=
(AlgEquiv.autCongr (adjoinRootXPowSubCEquiv hζ H (rootOfSplitsXPowSubC_pow a L)).symm).trans
(autAdjoinRootXPowSubCEquiv hζ H).symm | def | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autEquivRootsOfUnity | Suppose `L/K` is the splitting field of `Xⁿ - a`, then `Gal(L/K)` is isomorphic to the
roots of unity in `K` if `K` contains all of them.
Note that this does not depend on a choice of `ⁿ√a`. |
autEquivRootsOfUnity_apply_rootOfSplit [NeZero n] (σ : L ≃ₐ[K] L) :
σ (rootOfSplitsXPowSubC (NeZero.pos n) a L) =
autEquivRootsOfUnity hζ H L σ • (rootOfSplitsXPowSubC (NeZero.pos n) a L) := by
obtain ⟨η, rfl⟩ := (autEquivRootsOfUnity hζ H L).symm.surjective σ
rw [MulEquiv.apply_symm_apply, autEquivRootsOfUnity]
simp only [MulEquiv.symm_trans_apply, AlgEquiv.autCongr_symm, AlgEquiv.symm_symm,
MulEquiv.symm_symm, AlgEquiv.autCongr_apply, AlgEquiv.trans_apply,
adjoinRootXPowSubCEquiv_symm_eq_root, autAdjoinRootXPowSubCEquiv_root, map_smul,
adjoinRootXPowSubCEquiv_root]
rfl
include hα in | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autEquivRootsOfUnity_apply_rootOfSplit | null |
autEquivRootsOfUnity_smul [NeZero n] (σ : L ≃ₐ[K] L) :
autEquivRootsOfUnity hζ H L σ • α = σ α := by
have ⟨ζ, hζ'⟩ := hζ
have hn := NeZero.pos n
rw [mem_primitiveRoots hn] at hζ'
rw [← mem_nthRoots hn, (hζ'.map_of_injective (algebraMap K L).injective).nthRoots_eq
(rootOfSplitsXPowSubC_pow a L)] at hα
simp only [Multiset.mem_map, Multiset.mem_range] at hα
obtain ⟨i, _, rfl⟩ := hα
simp only [← map_pow, ← Algebra.smul_def, map_smul,
autEquivRootsOfUnity_apply_rootOfSplit hζ H L]
exact smul_comm _ _ _ | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autEquivRootsOfUnity_smul | null |
noncomputable
autEquivZmod [NeZero n] {ζ : K} (hζ : IsPrimitiveRoot ζ n) :
(L ≃ₐ[K] L) ≃* Multiplicative (ZMod n) :=
haveI hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)
(autEquivRootsOfUnity ⟨ζ, (mem_primitiveRoots hn).mpr hζ⟩ H L).trans
((MulEquiv.subgroupCongr (IsPrimitiveRoot.zpowers_eq
(hζ.isUnit_unit' hn)).symm).trans (AddEquiv.toMultiplicative'
(hζ.isUnit_unit' hn).zmodEquivZPowers.symm))
include hα in | def | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autEquivZmod | Suppose `L/K` is the splitting field of `Xⁿ - a`, and `ζ` is a `n`-th primitive root of unity
in `K`, then `Gal(L/K)` is isomorphic to `ZMod n`. |
autEquivZmod_symm_apply_intCast [NeZero n] {ζ : K} (hζ : IsPrimitiveRoot ζ n) (m : ℤ) :
(autEquivZmod H L hζ).symm (Multiplicative.ofAdd (m : ZMod n)) α = ζ ^ m • α := by
have hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)
rw [← autEquivRootsOfUnity_smul ⟨ζ, (mem_primitiveRoots hn).mpr hζ⟩ H L hα]
simp [MulEquiv.subgroupCongr_symm_apply, Subgroup.smul_def, Units.smul_def, autEquivZmod]
include hα in | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autEquivZmod_symm_apply_intCast | null |
autEquivZmod_symm_apply_natCast [NeZero n] {ζ : K} (hζ : IsPrimitiveRoot ζ n) (m : ℕ) :
(autEquivZmod H L hζ).symm (Multiplicative.ofAdd (m : ZMod n)) α = ζ ^ m • α := by
simpa only [Int.cast_natCast, zpow_natCast] using autEquivZmod_symm_apply_intCast H L hα hζ m
include hζ H in | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | autEquivZmod_symm_apply_natCast | null |
isCyclic_of_isSplittingField_X_pow_sub_C [NeZero n] : IsCyclic (L ≃ₐ[K] L) :=
have hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)
isCyclic_of_surjective _
(autEquivZmod H _ <| (mem_primitiveRoots hn).mp hζ.choose_spec).symm.surjective
include hζ H in | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | isCyclic_of_isSplittingField_X_pow_sub_C | null |
isGalois_of_isSplittingField_X_pow_sub_C : IsGalois K L :=
IsGalois.of_separable_splitting_field (separable_X_pow_sub_C_of_irreducible hζ a H)
include hζ H in | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | isGalois_of_isSplittingField_X_pow_sub_C | null |
finrank_of_isSplittingField_X_pow_sub_C : Module.finrank K L = n := by
have := Polynomial.IsSplittingField.finiteDimensional L (X ^ n - C a)
have := isGalois_of_isSplittingField_X_pow_sub_C hζ H L
have hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)
have : NeZero n := ⟨ne_zero_of_irreducible_X_pow_sub_C H⟩
rw [← IsGalois.card_aut_eq_finrank, Nat.card_congr ((autEquivZmod H L <|
(mem_primitiveRoots hn).mp hζ.choose_spec).toEquiv.trans Multiplicative.toAdd), Nat.card_zmod] | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | finrank_of_isSplittingField_X_pow_sub_C | null |
exists_root_adjoin_eq_top_of_isCyclic [IsGalois K L] [IsCyclic (L ≃ₐ[K] L)] :
∃ (α : L), α ^ (finrank K L) ∈ Set.range (algebraMap K L) ∧ K⟮α⟯ = ⊤ := by
have ⟨ζ, hζ⟩ := hK
rw [mem_primitiveRoots finrank_pos] at hζ
obtain ⟨σ, hσ⟩ := ‹IsCyclic (L ≃ₐ[K] L)›
have hσ' := orderOf_eq_card_of_forall_mem_zpowers hσ
have : IsRoot (minpoly K σ.toLinearMap) ζ := by
rw [IsGalois.card_aut_eq_finrank] at hσ'
simpa [minpoly_algEquiv_toLinearMap σ (isOfFinOrder_of_finite σ), hσ',
sub_eq_zero] using hζ.pow_eq_one
obtain ⟨v, hv⟩ := (Module.End.hasEigenvalue_of_isRoot this).exists_hasEigenvector
have hv' := hv.pow_apply
simp_rw [← AlgEquiv.pow_toLinearMap, AlgEquiv.toLinearMap_apply] at hv'
refine ⟨v, ?_, ?_⟩
· -- Since `v ^ n` is fixed by `σ` (`σ (v ^ n) = ζ ^ n • v ^ n = v ^ n`), it is in `K`.
rw [← IntermediateField.mem_bot,
← OrderIso.map_bot IsGalois.intermediateFieldEquivSubgroup.symm]
intro ⟨σ', hσ'⟩
obtain ⟨n, rfl : σ ^ n = σ'⟩ := mem_powers_iff_mem_zpowers.mpr (hσ σ')
rw [smul_pow', Submonoid.smul_def, AlgEquiv.smul_def, hv', smul_pow, ← pow_mul,
mul_comm, pow_mul, hζ.pow_eq_one, one_pow, one_smul]
· -- Since `σ` does not fix `K⟮α⟯`, `K⟮α⟯` is `L`.
apply IsGalois.intermediateFieldEquivSubgroup.injective
rw [map_top, eq_top_iff]
intro σ' hσ'
obtain ⟨n, rfl : σ ^ n = σ'⟩ := mem_powers_iff_mem_zpowers.mpr (hσ σ')
have := hσ' ⟨v, IntermediateField.mem_adjoin_simple_self K v⟩
simp only [AlgEquiv.smul_def, hv'] at this
conv_rhs at this => rw [← one_smul K v]
obtain ⟨k, rfl⟩ := hζ.dvd_of_pow_eq_one n (smul_left_injective K hv.2 this)
rw [pow_mul, ← IsGalois.card_aut_eq_finrank, pow_card_eq_one', one_pow]
exact one_mem _
variable {K L} | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | exists_root_adjoin_eq_top_of_isCyclic | If `L/K` is a cyclic extension of degree `n`, and `K` contains all `n`-th roots of unity,
then `L = K[α]` for some `α ^ n ∈ K`. |
irreducible_X_pow_sub_C_of_root_adjoin_eq_top
{a : K} {α : L} (ha : α ^ (finrank K L) = algebraMap K L a) (hα : K⟮α⟯ = ⊤) :
Irreducible (X ^ (finrank K L) - C a) := by
have : X ^ (finrank K L) - C a = minpoly K α := by
refine minpoly.unique _ _ (monic_X_pow_sub_C _ finrank_pos.ne.symm) ?_ ?_
· simp only [aeval_def, eval₂_sub, eval₂_X_pow, ha, eval₂_C, sub_self]
· intro q hq hq'
refine le_trans ?_ (degree_le_of_dvd (minpoly.dvd _ _ hq') hq.ne_zero)
rw [degree_X_pow_sub_C finrank_pos,
degree_eq_natDegree (minpoly.ne_zero (IsIntegral.of_finite K α)),
← IntermediateField.adjoin.finrank (IsIntegral.of_finite K α), hα, Nat.cast_le]
exact (finrank_top K L).ge
exact this ▸ minpoly.irreducible (IsIntegral.of_finite K α)
include hK in | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | irreducible_X_pow_sub_C_of_root_adjoin_eq_top | null |
isSplittingField_X_pow_sub_C_of_root_adjoin_eq_top
{a : K} {α : L} (ha : α ^ (finrank K L) = algebraMap K L a) (hα : K⟮α⟯ = ⊤) :
IsSplittingField K L (X ^ (finrank K L) - C a) := by
constructor
· rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_C,
Polynomial.map_X]
have ⟨_, hζ⟩ := hK
rw [mem_primitiveRoots finrank_pos] at hζ
exact X_pow_sub_C_splits_of_isPrimitiveRoot (hζ.map_of_injective (algebraMap K _).injective) ha
· rw [eq_top_iff, ← IntermediateField.top_toSubalgebra, ← hα,
IntermediateField.adjoin_simple_toSubalgebra_of_integral (IsIntegral.of_finite K α)]
apply Algebra.adjoin_mono
rw [Set.singleton_subset_iff, mem_rootSet_of_ne (X_pow_sub_C_ne_zero finrank_pos a),
aeval_def, eval₂_sub, eval₂_X_pow, eval₂_C, ha, sub_self] | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | isSplittingField_X_pow_sub_C_of_root_adjoin_eq_top | null |
isCyclic_tfae (K L) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L]
(hK : (primitiveRoots (Module.finrank K L) K).Nonempty) :
List.TFAE [
IsGalois K L ∧ IsCyclic (L ≃ₐ[K] L),
∃ a : K, Irreducible (X ^ (finrank K L) - C a) ∧
IsSplittingField K L (X ^ (finrank K L) - C a),
∃ (α : L), α ^ (finrank K L) ∈ Set.range (algebraMap K L) ∧ K⟮α⟯ = ⊤] := by
have : NeZero (Module.finrank K L) := NeZero.of_pos finrank_pos
tfae_have 1 → 3
| ⟨inst₁, inst₂⟩ => exists_root_adjoin_eq_top_of_isCyclic K L hK
tfae_have 3 → 2
| ⟨α, ⟨a, ha⟩, hα⟩ => ⟨a, irreducible_X_pow_sub_C_of_root_adjoin_eq_top ha.symm hα,
isSplittingField_X_pow_sub_C_of_root_adjoin_eq_top hK ha.symm hα⟩
tfae_have 2 → 1
| ⟨a, H, inst⟩ => ⟨isGalois_of_isSplittingField_X_pow_sub_C hK H L,
isCyclic_of_isSplittingField_X_pow_sub_C hK H L⟩
tfae_finish | lemma | FieldTheory | [
"Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots",
"Mathlib.FieldTheory.Galois.Basic",
"Mathlib.FieldTheory.KummerPolynomial",
"Mathlib.LinearAlgebra.Eigenspace.Minpoly",
"Mathlib.RingTheory.Norm.Basic"
] | Mathlib/FieldTheory/KummerExtension.lean | isCyclic_tfae | Suppose `L/K` is a finite extension of dimension `n`, and `K` contains all `n`-th roots of unity.
Then `L/K` is cyclic iff
`L` is a splitting field of some irreducible polynomial of the form `Xⁿ - a : K[X]` iff
`L = K[α]` for some `αⁿ ∈ K`. |
root_X_pow_sub_C_pow (n : ℕ) (a : K) :
(AdjoinRoot.root (X ^ n - C a)) ^ n = AdjoinRoot.of _ a := by
rw [← sub_eq_zero, ← AdjoinRoot.eval₂_root, eval₂_sub, eval₂_C, eval₂_pow, eval₂_X] | lemma | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | root_X_pow_sub_C_pow | null |
root_X_pow_sub_C_ne_zero {n : ℕ} (hn : 1 < n) (a : K) :
(AdjoinRoot.root (X ^ n - C a)) ≠ 0 :=
mk_ne_zero_of_natDegree_lt (monic_X_pow_sub_C _ (Nat.ne_zero_of_lt hn))
X_ne_zero <| by rwa [natDegree_X_pow_sub_C, natDegree_X] | lemma | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | root_X_pow_sub_C_ne_zero | null |
root_X_pow_sub_C_ne_zero' {n : ℕ} {a : K} (hn : 0 < n) (ha : a ≠ 0) :
(AdjoinRoot.root (X ^ n - C a)) ≠ 0 := by
obtain (rfl | hn) := (Nat.succ_le_iff.mpr hn).eq_or_lt
· rw [pow_one]
intro e
refine mk_ne_zero_of_natDegree_lt (monic_X_sub_C a) (C_ne_zero.mpr ha) (by simp) ?_
trans AdjoinRoot.mk (X - C a) (X - (X - C a))
· rw [sub_sub_cancel]
· rw [map_sub, mk_self, sub_zero, mk_X, e]
· exact root_X_pow_sub_C_ne_zero hn a | lemma | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | root_X_pow_sub_C_ne_zero' | null |
ne_zero_of_irreducible_X_pow_sub_C {n : ℕ} {a : K} (H : Irreducible (X ^ n - C a)) :
n ≠ 0 := by
rintro rfl
rw [pow_zero, ← C.map_one, ← map_sub] at H
exact not_irreducible_C _ H | lemma | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | ne_zero_of_irreducible_X_pow_sub_C | null |
ne_zero_of_irreducible_X_pow_sub_C' {n : ℕ} (hn : n ≠ 1) {a : K}
(H : Irreducible (X ^ n - C a)) : a ≠ 0 := by
rintro rfl
rw [map_zero, sub_zero] at H
exact not_irreducible_pow hn H | lemma | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | ne_zero_of_irreducible_X_pow_sub_C' | null |
root_X_pow_sub_C_eq_zero_iff {n : ℕ} {a : K} (H : Irreducible (X ^ n - C a)) :
(AdjoinRoot.root (X ^ n - C a)) = 0 ↔ a = 0 := by
have hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)
refine ⟨not_imp_not.mp (root_X_pow_sub_C_ne_zero' hn), ?_⟩
rintro rfl
have := not_imp_not.mp (fun hn ↦ ne_zero_of_irreducible_X_pow_sub_C' hn H) rfl
rw [this, pow_one, map_zero, sub_zero, ← mk_X, mk_self] | lemma | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | root_X_pow_sub_C_eq_zero_iff | null |
root_X_pow_sub_C_ne_zero_iff {n : ℕ} {a : K} (H : Irreducible (X ^ n - C a)) :
(AdjoinRoot.root (X ^ n - C a)) ≠ 0 ↔ a ≠ 0 :=
(root_X_pow_sub_C_eq_zero_iff H).not | lemma | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | root_X_pow_sub_C_ne_zero_iff | null |
pow_ne_of_irreducible_X_pow_sub_C {n : ℕ} {a : K}
(H : Irreducible (X ^ n - C a)) {m : ℕ} (hm : m ∣ n) (hm' : m ≠ 1) (b : K) : b ^ m ≠ a := by
have hn : n ≠ 0 := fun e ↦ not_irreducible_C
(1 - a) (by simpa only [e, pow_zero, ← C.map_one, ← map_sub] using H)
obtain ⟨k, rfl⟩ := hm
rintro rfl
obtain ⟨q, hq⟩ := sub_dvd_pow_sub_pow (X ^ k) (C b) m
rw [mul_comm, pow_mul, map_pow, hq] at H
have : degree q = 0 := by
simpa [isUnit_iff_degree_eq_zero, degree_X_pow_sub_C,
Nat.pos_iff_ne_zero, (mul_ne_zero_iff.mp hn).2] using H.2 rfl
apply_fun degree at hq
simp only [this, ← pow_mul, mul_comm k m, degree_X_pow_sub_C, Nat.pos_iff_ne_zero.mpr hn,
Nat.pos_iff_ne_zero.mpr (mul_ne_zero_iff.mp hn).2, degree_mul, ← map_pow, add_zero,
Nat.cast_injective.eq_iff] at hq
exact hm' ((mul_eq_right₀ (mul_ne_zero_iff.mp hn).2).mp hq) | theorem | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | pow_ne_of_irreducible_X_pow_sub_C | null |
@[stacks 09HF "We proved the result without the condition that `K` is char p in 09HF."]
X_pow_sub_C_irreducible_of_prime {p : ℕ} (hp : p.Prime) {a : K} (ha : ∀ b : K, b ^ p ≠ a) :
Irreducible (X ^ p - C a) := by
have : ¬ IsUnit (X ^ p - C a) := by
rw [Polynomial.isUnit_iff_degree_eq_zero, degree_X_pow_sub_C hp.pos, Nat.cast_eq_zero]
exact hp.ne_zero
have ⟨g, hg, hg'⟩ := WfDvdMonoid.exists_irreducible_factor this (X_pow_sub_C_ne_zero hp.pos a)
suffices natDegree g = p from (associated_of_dvd_of_natDegree_le hg'
(X_pow_sub_C_ne_zero hp.pos a) (this.trans natDegree_X_pow_sub_C.symm).ge).irreducible hg
by_contra h
have key : (Algebra.norm K (AdjoinRoot.root g)) ^ p = a ^ g.natDegree := by
have := eval₂_eq_zero_of_dvd_of_eval₂_eq_zero _ _ hg' (AdjoinRoot.eval₂_root g)
rw [eval₂_sub, eval₂_pow, eval₂_C, eval₂_X, sub_eq_zero] at this
rw [← map_pow, this, ← AdjoinRoot.algebraMap_eq, Algebra.norm_algebraMap,
(powerBasis hg.ne_zero).finrank, powerBasis_dim hg.ne_zero]
have : p.Coprime (natDegree g) := hp.coprime_iff_not_dvd.mpr (fun e ↦ h (((natDegree_le_of_dvd hg'
(X_pow_sub_C_ne_zero hp.pos a)).trans_eq natDegree_X_pow_sub_C).antisymm (Nat.le_of_dvd
(natDegree_pos_iff_degree_pos.mpr <| Polynomial.degree_pos_of_irreducible hg) e)))
exact ha _ ((pow_mem_range_pow_of_coprime this.symm a).mp ⟨_, key⟩).choose_spec | theorem | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | X_pow_sub_C_irreducible_of_prime | Let `p` be a prime number. Let `K` be a field.
Let `t ∈ K` be an element which does not have a `p`th root in `K`.
Then the polynomial `x ^ p - t` is irreducible over `K`. |
X_pow_sub_C_irreducible_iff_of_prime {p : ℕ} (hp : p.Prime) {a : K} :
Irreducible (X ^ p - C a) ↔ ∀ b, b ^ p ≠ a :=
⟨(pow_ne_of_irreducible_X_pow_sub_C · dvd_rfl hp.ne_one), X_pow_sub_C_irreducible_of_prime hp⟩ | theorem | FieldTheory | [
"Mathlib.RingTheory.AdjoinRoot",
"Mathlib.RingTheory.Norm.Defs"
] | Mathlib/FieldTheory/KummerPolynomial.lean | X_pow_sub_C_irreducible_iff_of_prime | null |
taylor_mem_nonZeroDivisors (hp : p ∈ R[X]⁰) : taylor r p ∈ R[X]⁰ := by
rw [mem_nonZeroDivisors_iff_right]
intro x hx
have : x = taylor (r - r) x := by simp
rwa [this, sub_eq_add_neg, ← taylor_taylor, ← taylor_mul,
LinearMap.map_eq_zero_iff _ (taylor_injective _), mul_right_mem_nonZeroDivisors_eq_zero_iff hp,
LinearMap.map_eq_zero_iff _ (taylor_injective _)] at hx | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | taylor_mem_nonZeroDivisors | null |
laurentAux : RatFunc R →+* RatFunc R :=
RatFunc.mapRingHom
( { toFun := taylor r
map_add' := map_add (taylor r)
map_mul' := taylor_mul _
map_zero' := map_zero (taylor r)
map_one' := taylor_one r } : R[X] →+* R[X])
(taylor_mem_nonZeroDivisors _) | def | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurentAux | The Laurent expansion of rational functions about a value.
Auxiliary definition, usage when over integral domains should prefer `RatFunc.laurent`. |
laurentAux_ofFractionRing_mk (q : R[X]⁰) :
laurentAux r (ofFractionRing (Localization.mk p q)) =
ofFractionRing (.mk (taylor r p) ⟨taylor r q, taylor_mem_nonZeroDivisors r q q.prop⟩) :=
map_apply_ofFractionRing_mk _ _ _ _
variable [IsDomain R] | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurentAux_ofFractionRing_mk | null |
laurentAux_div :
laurentAux r (algebraMap _ _ p / algebraMap _ _ q) =
algebraMap _ _ (taylor r p) / algebraMap _ _ (taylor r q) :=
map_apply_div _ (by exact taylor_mem_nonZeroDivisors r) _ _
@[simp] | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurentAux_div | null |
laurentAux_algebraMap : laurentAux r (algebraMap _ _ p) = algebraMap _ _ (taylor r p) := by
rw [← mk_one, ← mk_one, mk_eq_div, laurentAux_div, mk_eq_div, taylor_one, map_one, map_one] | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurentAux_algebraMap | null |
laurent : RatFunc R →ₐ[R] RatFunc R :=
RatFunc.mapAlgHom (.ofLinearMap (taylor r) (taylor_one _) (taylor_mul _))
(taylor_mem_nonZeroDivisors _) | def | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurent | The Laurent expansion of rational functions about a value. |
laurent_div :
laurent r (algebraMap _ _ p / algebraMap _ _ q) =
algebraMap _ _ (taylor r p) / algebraMap _ _ (taylor r q) :=
laurentAux_div r p q
@[simp] | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurent_div | null |
laurent_algebraMap : laurent r (algebraMap _ _ p) = algebraMap _ _ (taylor r p) :=
laurentAux_algebraMap _ _
@[simp] | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurent_algebraMap | null |
laurent_X : laurent r X = X + C r := by
rw [← algebraMap_X, laurent_algebraMap, taylor_X, map_add, algebraMap_C]
@[simp] | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurent_X | null |
laurent_C (x : R) : laurent r (C x) = C x := by
rw [← algebraMap_C, laurent_algebraMap, taylor_C]
@[simp] | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurent_C | null |
laurent_at_zero : laurent 0 f = f := by induction f using RatFunc.induction_on; simp | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurent_at_zero | null |
laurent_laurent : laurent r (laurent s f) = laurent (r + s) f := by
induction f using RatFunc.induction_on
simp_rw [laurent_div, taylor_taylor] | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurent_laurent | null |
laurent_injective : Function.Injective (laurent r) := fun _ _ h => by
simpa [laurent_laurent] using congr_arg (laurent (-r)) h | theorem | FieldTheory | [
"Mathlib.Algebra.Polynomial.Taylor",
"Mathlib.FieldTheory.RatFunc.AsPolynomial"
] | Mathlib/FieldTheory/Laurent.lean | laurent_injective | null |
protected LinearDisjoint : Prop :=
A.toSubalgebra.LinearDisjoint (IsScalarTower.toAlgHom F L E).range | abbrev | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | LinearDisjoint | If `A` is an intermediate field of `E / F`, and `E / L / F` is a field extension tower,
then `A` and `L` are linearly disjoint, if they are linearly disjoint as subalgebras of `E`
(`Subalgebra.LinearDisjoint`). |
linearDisjoint_iff :
A.LinearDisjoint L ↔ A.toSubalgebra.LinearDisjoint (IsScalarTower.toAlgHom F L E).range :=
Iff.rfl
variable {A B L} | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | linearDisjoint_iff | null |
linearDisjoint_iff' :
A.LinearDisjoint B ↔ A.toSubalgebra.LinearDisjoint B.toSubalgebra := by
rw [linearDisjoint_iff]
congr!
ext; simp | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | linearDisjoint_iff' | Two intermediate fields are linearly disjoint if and only if
they are linearly disjoint as subalgebras. |
LinearDisjoint.symm (H : A.LinearDisjoint B) : B.LinearDisjoint A :=
linearDisjoint_iff'.2 (linearDisjoint_iff'.1 H).symm | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | LinearDisjoint.symm | Linear disjointness is symmetric. |
linearDisjoint_comm : A.LinearDisjoint B ↔ B.LinearDisjoint A :=
⟨LinearDisjoint.symm, LinearDisjoint.symm⟩ | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | linearDisjoint_comm | Linear disjointness is symmetric. |
LinearDisjoint.symm' (H : (IsScalarTower.toAlgHom F L E).fieldRange.LinearDisjoint L') :
(IsScalarTower.toAlgHom F L' E).fieldRange.LinearDisjoint L :=
Subalgebra.LinearDisjoint.symm H | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | LinearDisjoint.symm' | Linear disjointness is symmetric. |
linearDisjoint_comm' :
(IsScalarTower.toAlgHom F L E).fieldRange.LinearDisjoint L' ↔
(IsScalarTower.toAlgHom F L' E).fieldRange.LinearDisjoint L :=
⟨LinearDisjoint.symm', LinearDisjoint.symm'⟩ | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | linearDisjoint_comm' | Linear disjointness is symmetric. |
map (H : A.LinearDisjoint B) {K : Type*} [Field K] [Algebra F K]
(f : E →ₐ[F] K) : (A.map f).LinearDisjoint (B.map f) :=
linearDisjoint_iff'.2 ((linearDisjoint_iff'.1 H).map f f.injective) | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | map | Linear disjointness of intermediate fields is preserved by algebra homomorphisms. |
map' (H : A.LinearDisjoint L) (K : Type*) [Field K] [Algebra F K] [Algebra L K]
[IsScalarTower F L K] [Algebra E K] [IsScalarTower F E K] [IsScalarTower L E K] :
(A.map (IsScalarTower.toAlgHom F E K)).LinearDisjoint L := by
rw [linearDisjoint_iff] at H ⊢
have := H.map (IsScalarTower.toAlgHom F E K) (RingHom.injective _)
rw [← AlgHom.range_comp] at this
convert this
ext; exact IsScalarTower.algebraMap_apply L E K _ | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | map' | Linear disjointness of an intermediate field with a tower of field embeddings is preserved by
algebra homomorphisms. |
map'' {L' : Type*} [Field L'] [Algebra F L'] [Algebra L' E] [IsScalarTower F L' E]
(H : (IsScalarTower.toAlgHom F L E).fieldRange.LinearDisjoint L')
(K : Type*) [Field K] [Algebra F K] [Algebra L K] [IsScalarTower F L K]
[Algebra L' K] [IsScalarTower F L' K] [Algebra E K] [IsScalarTower F E K]
[IsScalarTower L E K] [IsScalarTower L' E K] :
(IsScalarTower.toAlgHom F L K).fieldRange.LinearDisjoint L' := by
rw [linearDisjoint_iff] at H ⊢
have := H.map (IsScalarTower.toAlgHom F E K) (RingHom.injective _)
simp_rw [AlgHom.fieldRange_toSubalgebra, ← AlgHom.range_comp] at this
rw [AlgHom.fieldRange_toSubalgebra]
convert this <;> (ext; exact IsScalarTower.algebraMap_apply _ E K _)
variable (A) in | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | map'' | Linear disjointness is preserved by algebra homomorphism. |
self_right : A.LinearDisjoint F := Subalgebra.LinearDisjoint.bot_right _
variable (A) in | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | self_right | null |
bot_right : A.LinearDisjoint (⊥ : IntermediateField F E) :=
linearDisjoint_iff'.2 (Subalgebra.LinearDisjoint.bot_right _)
variable (F E L) in | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | bot_right | null |
bot_left : (⊥ : IntermediateField F E).LinearDisjoint L :=
Subalgebra.LinearDisjoint.bot_left _ | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | bot_left | null |
linearIndependent_left (H : A.LinearDisjoint L)
{ι : Type*} {a : ι → A} (ha : LinearIndependent F a) : LinearIndependent L (A.val ∘ a) :=
(Subalgebra.LinearDisjoint.linearIndependent_left_of_flat H ha).map_of_injective_injective
(AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F L E)) (AddMonoidHom.id E)
(by simp) (by simp) (fun _ _ ↦ by simp_rw [Algebra.smul_def]; rfl) | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | linearIndependent_left | If `A` and `L` are linearly disjoint, then any `F`-linearly independent family on `A` remains
linearly independent over `L`. |
of_basis_left {ι : Type*} (a : Basis ι F A)
(H : LinearIndependent L (A.val ∘ a)) : A.LinearDisjoint L :=
Subalgebra.LinearDisjoint.of_basis_left _ _ a <| H.map_of_surjective_injective
(AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F L E)) (AddMonoidHom.id E)
(AlgEquiv.surjective _) (by simp) (fun _ _ ↦ by simp_rw [Algebra.smul_def]; rfl) | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | of_basis_left | If there exists an `F`-basis of `A` which remains linearly independent over `L`, then
`A` and `L` are linearly disjoint. |
linearIndependent_right (H : A.LinearDisjoint B)
{ι : Type*} {b : ι → B} (hb : LinearIndependent F b) : LinearIndependent A (B.val ∘ b) :=
(linearDisjoint_iff'.1 H).linearIndependent_right_of_flat hb | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | linearIndependent_right | If `A` and `B` are linearly disjoint, then any `F`-linearly independent family on `B` remains
linearly independent over `A`. |
noncomputable basisOfBasisRight (H : A.LinearDisjoint B)
(H' : A.toSubalgebra ⊔ B.toSubalgebra = ⊤) {ι : Type*} (b : Basis ι F B) :
Basis ι A E :=
(linearDisjoint_iff'.mp H).basisOfBasisRight H' b
@[simp] | def | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | basisOfBasisRight | If `A` and `B` are linearly disjoint and such that `A.toSubalgebra ⊔ B.toSubalgebra = ⊤`,
then any `F`-basis of `B` is also an `A`-basis of `E`.
Note that the condition `A.toSubalgebra ⊔ B.toSubalgebra = ⊤` is equivalent to
`A ⊔ B = ⊤` in many cases, see `IntermediateField.sup_toSubalgebra_of_isAlgebraic_right` and similar
results. |
basisOfBasisRight_apply (H : A.LinearDisjoint B) (H' : A.toSubalgebra ⊔ B.toSubalgebra = ⊤)
{ι : Type*} (b : Basis ι F B) (i : ι) :
H.basisOfBasisRight H' b i = algebraMap B E (b i) :=
(linearDisjoint_iff'.mp H).algebraMap_basisOfBasisRight_apply H' b i | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | basisOfBasisRight_apply | null |
algebraMap_basisOfBasisRight_repr_apply (H : A.LinearDisjoint B)
(H' : A.toSubalgebra ⊔ B.toSubalgebra = ⊤) {ι : Type*} (b : Basis ι F B) (x : B) (i : ι) :
algebraMap A E ((H.basisOfBasisRight H' b).repr x i) = algebraMap F E (b.repr x i) :=
(linearDisjoint_iff'.mp H).algebraMap_basisOfBasisRight_repr_apply H' b x i | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | algebraMap_basisOfBasisRight_repr_apply | null |
noncomputable basisOfBasisLeft (H : A.LinearDisjoint B)
(H' : A.toSubalgebra ⊔ B.toSubalgebra = ⊤) {ι : Type*} (b : Basis ι F A) :
Basis ι B E :=
(linearDisjoint_iff'.mp H).basisOfBasisLeft H' b
@[simp] | def | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | basisOfBasisLeft | If `A` and `B` are linearly disjoint and such that `A.toSubalgebra ⊔ B.toSubalgebra = ⊤`,
then any `F`-basis of `A` is also a `B`-basis of `E`.
Note that the condition `A.toSubalgebra ⊔ B.toSubalgebra = ⊤` is equivalent to
`A ⊔ B = ⊤` in many cases, see `IntermediateField.sup_toSubalgebra_of_isAlgebraic_right` and similar
results. |
basisOfBasisLeft_apply (H : A.LinearDisjoint B) (H' : A.toSubalgebra ⊔ B.toSubalgebra = ⊤)
{ι : Type*} (b : Basis ι F A) (i : ι) :
H.basisOfBasisLeft H' b i = algebraMap A E (b i) :=
(linearDisjoint_iff'.mp H).basisOfBasisLeft_apply H' b i | theorem | FieldTheory | [
"Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality",
"Mathlib.RingTheory.LinearDisjoint"
] | Mathlib/FieldTheory/LinearDisjoint.lean | basisOfBasisLeft_apply | null |
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