| import Mathlib | |
| namespace OAI | |
| /-! Finite fields containing radicals of all algebraic integer units. -/ | |
| noncomputable section | |
| open NumberField | |
| namespace IntegralCharacterVarieties.Arithmetic | |
| def unitRootExtension (F : Type*) [Field F] [Algebra F ℂ] | |
| (ζ : ℂ) (v : (𝓞 F)ˣ → ℂ) : IntermediateField F ℂ := | |
| IntermediateField.adjoin F (insert ζ (Set.range v)) | |
| theorem root_of_unit_is_ringOfIntegers_unit | |
| {F E : Type*} [Field F] [Field E] [Algebra F E] | |
| {r : ℕ} (hr : 0 < r) (u : (𝓞 F)ˣ) {x : E} | |
| (hx : x ^ r = algebraMap F E (u : F)) : | |
| ∃ w : (𝓞 E)ˣ, (w : E) = x := by | |
| have hint : IsIntegral ℤ x := by | |
| apply IsIntegral.of_pow hr | |
| rw [hx] | |
| exact map_isIntegral_int (algebraMap F E) | |
| (RingOfIntegers.isIntegral_coe (u : 𝓞 F)) | |
| let y : 𝓞 E := ⟨x, hint⟩ | |
| have hpow : y ^ r = RingOfIntegers.mapRingHom (algebraMap F E) (u : 𝓞 F) := by | |
| apply RingOfIntegers.ext | |
| change x ^ r = algebraMap F E (u : F) | |
| exact hx | |
| have hy : IsUnit (y ^ r) := by | |
| rw [hpow] | |
| exact u.isUnit.map (RingOfIntegers.mapRingHom (algebraMap F E)) | |
| obtain ⟨w, hw⟩ := (isUnit_pow_iff hr.ne').mp hy | |
| refine ⟨w, ?_⟩ | |
| rw [hw] | |
| rfl | |
| theorem finiteDimensional_unitRootExtension_and_roots | |
| (F : Type*) [Field F] [NumberField F] [Algebra F ℂ] | |
| {r : ℕ} (hr : 0 < r) (ζ : ℂ) (hζ : IsPrimitiveRoot ζ r) | |
| (v : (𝓞 F)ˣ → ℂ) (hv : ∀ u, v u ^ r = algebraMap F ℂ (u : F)) : | |
| FiniteDimensional F (unitRootExtension F ζ v) ∧ | |
| ∀ (u : (𝓞 F)ˣ) (x : ℂ), x ^ r = algebraMap F ℂ (u : F) → | |
| x ∈ unitRootExtension F ζ v := by | |
| classical | |
| let : NeZero r := ⟨hr.ne'⟩ | |
| obtain ⟨S, hSgen, hSfin⟩ := | |
| Monoid.fg_iff.mp (inferInstance : Monoid.FG (𝓞 F)ˣ) | |
| let E : IntermediateField F ℂ := IntermediateField.adjoin F (insert ζ (v '' S)) | |
| have hζE : ζ ∈ E := IntermediateField.subset_adjoin F _ (Set.mem_insert _ _) | |
| let H : Submonoid (𝓞 F)ˣ := | |
| { carrier := {u | ∃ a : ℂ, a ∈ E ∧ a ^ r = algebraMap F ℂ (u : F)} | |
| one_mem' := by | |
| refine ⟨1, E.one_mem, ?_⟩ | |
| simp | |
| mul_mem' := by | |
| rintro u w ⟨a, ha, har⟩ ⟨b, hb, hbr⟩ | |
| refine ⟨a * b, E.mul_mem ha hb, ?_⟩ | |
| simp [mul_pow, har, hbr] } | |
| have hSH : S ⊆ H := by | |
| intro u hu | |
| exact ⟨v u, IntermediateField.subset_adjoin F _ | |
| (Set.mem_insert_of_mem _ ⟨u, hu, rfl⟩), hv u⟩ | |
| have hH : H = ⊤ := by | |
| apply top_unique | |
| rw [← hSgen] | |
| exact Submonoid.closure_le.mpr hSH | |
| have hroots : ∀ (u : (𝓞 F)ˣ) (x : ℂ), | |
| x ^ r = algebraMap F ℂ (u : F) → x ∈ E := by | |
| intro u x hx | |
| have hu : u ∈ H := by rw [hH]; trivial | |
| obtain ⟨a, ha, har⟩ := hu | |
| have humap : algebraMap F ℂ (u : F) ≠ 0 := | |
| (map_ne_zero (algebraMap F ℂ)).mpr (NumberField.Units.coe_ne_zero u) | |
| have ha0 : a ≠ 0 := by | |
| intro h | |
| apply humap | |
| rw [← har, h, zero_pow hr.ne'] | |
| have hquot : (x / a) ^ r = 1 := by | |
| rw [div_pow, hx, har, div_self humap] | |
| obtain ⟨i, _, hi⟩ := hζ.eq_pow_of_pow_eq_one hquot | |
| have hxa : x = ζ ^ i * a := (div_eq_iff ha0).mp hi.symm | |
| rw [hxa] | |
| exact E.mul_mem (E.pow_mem hζE i) ha | |
| have hLE : unitRootExtension F ζ v ≤ E := by | |
| apply IntermediateField.adjoin_le_iff.mpr | |
| intro x hx | |
| rcases hx with rfl | ⟨u, rfl⟩ | |
| · exact hζE | |
| · exact hroots u (v u) (hv u) | |
| have hEL : E ≤ unitRootExtension F ζ v := by | |
| apply IntermediateField.adjoin.mono | |
| intro x hx | |
| rcases hx with rfl | ⟨u, _, rfl⟩ | |
| · exact Set.mem_insert _ _ | |
| · exact Set.mem_insert_of_mem _ ⟨u, rfl⟩ | |
| have hEq : unitRootExtension F ζ v = E := le_antisymm hLE hEL | |
| constructor | |
| · rw [hEq] | |
| let : Fintype ↥(insert ζ (v '' S)) := ((hSfin.image v).insert ζ).fintype | |
| apply IntermediateField.finiteDimensional_adjoin | |
| intro x hx | |
| rcases hx with rfl | ⟨u, _, rfl⟩ | |
| · apply IsIntegral.of_pow hr | |
| rw [hζ.pow_eq_one] | |
| exact isIntegral_one | |
| · apply IsIntegral.of_pow hr | |
| rw [hv] | |
| exact isIntegral_algebraMap | |
| · intro u x hx | |
| rw [hEq] | |
| exact hroots u x hx | |
| theorem uniform_field | |
| (F : Type*) [Field F] [NumberField F] [Algebra F ℂ] | |
| {r : ℕ} (hr : 0 < r) (ζ : ℂ) (hζ : IsPrimitiveRoot ζ r) | |
| (v : (𝓞 F)ˣ → ℂ) (hv : ∀ u, v u ^ r = algebraMap F ℂ (u : F)) : | |
| FiniteDimensional F (unitRootExtension F ζ v) ∧ | |
| ∀ (u : (𝓞 F)ˣ) (x : ℂ), x ^ r = algebraMap F ℂ (u : F) → | |
| ∃ w : (𝓞 (unitRootExtension F ζ v))ˣ, | |
| algebraMap (unitRootExtension F ζ v) ℂ | |
| (w : unitRootExtension F ζ v) = x := by | |
| obtain ⟨hfin, hroots⟩ := finiteDimensional_unitRootExtension_and_roots F hr ζ hζ v hv | |
| refine ⟨hfin, ?_⟩ | |
| intro u x hx | |
| let y : unitRootExtension F ζ v := ⟨x, hroots u x hx⟩ | |
| have hy : y ^ r = algebraMap F (unitRootExtension F ζ v) (u : F) := by | |
| apply Subtype.ext | |
| change x ^ r = algebraMap F ℂ (u : F) | |
| exact hx | |
| obtain ⟨w, hw⟩ := root_of_unit_is_ringOfIntegers_unit hr u hy | |
| refine ⟨w, ?_⟩ | |
| rw [hw] | |
| rfl | |
| def primitiveRoot (n : ℕ) : ℂ := Complex.exp (2 * Real.pi * Complex.I / n) | |
| theorem primitiveRoot_isPrimitiveRoot {n : ℕ} (hn : 0 < n) : | |
| IsPrimitiveRoot (primitiveRoot n) n := | |
| Complex.isPrimitiveRoot_exp n hn.ne' | |
| def unitRootChoice (F : Type*) [Field F] [Algebra F ℂ] | |
| (r : ℕ) (hr : 0 < r) (u : (𝓞 F)ˣ) : ℂ := | |
| Classical.choose (IsAlgClosed.exists_pow_nat_eq (algebraMap F ℂ (u : F)) hr) | |
| theorem unitRootChoice_pow (F : Type*) [Field F] [Algebra F ℂ] | |
| (r : ℕ) (hr : 0 < r) (u : (𝓞 F)ˣ) : | |
| unitRootChoice F r hr u ^ r = algebraMap F ℂ (u : F) := | |
| Classical.choose_spec (IsAlgClosed.exists_pow_nat_eq (algebraMap F ℂ (u : F)) hr) | |
| def radicalField (F : Type*) [Field F] [Algebra F ℂ] | |
| (r : ℕ) (hr : 0 < r) : IntermediateField F ℂ := | |
| unitRootExtension F (primitiveRoot r) (unitRootChoice F r hr) | |
| theorem radicalField_finite_and_units | |
| (F : Type*) [Field F] [NumberField F] [Algebra F ℂ] | |
| {r : ℕ} (hr : 0 < r) : | |
| FiniteDimensional F (radicalField F r hr) ∧ | |
| ∀ (u : (𝓞 F)ˣ) (x : ℂ), x ^ r = algebraMap F ℂ (u : F) → | |
| ∃ w : (𝓞 (radicalField F r hr))ˣ, | |
| algebraMap (radicalField F r hr) ℂ (w : radicalField F r hr) = x := | |
| uniform_field F hr (primitiveRoot r) (primitiveRoot_isPrimitiveRoot hr) | |
| (unitRootChoice F r hr) (unitRootChoice_pow F r hr) | |
| def auxiliaryField (K : Type*) [Field K] [Algebra K ℂ] | |
| (p q : ℕ) : IntermediateField K ℂ := | |
| IntermediateField.adjoin K | |
| {((Real.sqrt 2 : ℝ) : ℂ), primitiveRoot (p * q)} | |
| theorem auxiliaryField_finite | |
| (K : Type*) [Field K] [Algebra K ℂ] | |
| {p q : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) : | |
| FiniteDimensional K (auxiliaryField K p q) := by | |
| apply IntermediateField.finiteDimensional_adjoin | |
| intro x hx | |
| rcases hx with rfl | rfl | |
| · apply IsIntegral.of_pow (n := 2) (by norm_num) | |
| have hsqrt : ((Real.sqrt 2 : ℝ) : ℂ) ^ 2 = algebraMap K ℂ (2 : K) := by | |
| rw [map_ofNat] | |
| norm_cast | |
| norm_num [Real.sq_sqrt] | |
| rw [hsqrt] | |
| exact isIntegral_algebraMap | |
| · apply IsIntegral.of_pow (Nat.mul_pos hp.pos hq.pos) | |
| rw [(primitiveRoot_isPrimitiveRoot (Nat.mul_pos hp.pos hq.pos)).pow_eq_one] | |
| exact isIntegral_one | |
| theorem specified_field_finite_and_units | |
| (K : Type*) [Field K] [NumberField K] [Algebra K ℂ] | |
| {r p q : ℕ} (hr : 0 < r) (hp : Nat.Prime p) (hq : Nat.Prime q) : | |
| FiniteDimensional K (radicalField (auxiliaryField K p q) r hr) ∧ | |
| ∀ (u : (𝓞 (auxiliaryField K p q))ˣ) (x : ℂ), | |
| x ^ r = algebraMap (auxiliaryField K p q) ℂ (u : auxiliaryField K p q) → | |
| ∃ w : (𝓞 (radicalField (auxiliaryField K p q) r hr))ˣ, | |
| algebraMap (radicalField (auxiliaryField K p q) r hr) ℂ | |
| (w : radicalField (auxiliaryField K p q) r hr) = x := by | |
| let : FiniteDimensional K (auxiliaryField K p q) := auxiliaryField_finite K hp hq | |
| let : NumberField (auxiliaryField K p q) := | |
| NumberField.of_module_finite K (auxiliaryField K p q) | |
| obtain ⟨hfin, hunits⟩ := radicalField_finite_and_units (auxiliaryField K p q) hr | |
| let : FiniteDimensional (auxiliaryField K p q) | |
| (radicalField (auxiliaryField K p q) r hr) := hfin | |
| exact ⟨FiniteDimensional.trans K (auxiliaryField K p q) | |
| (radicalField (auxiliaryField K p q) r hr), hunits⟩ | |
| theorem exists_distinct_primes_gt (r : ℕ) : | |
| ∃ p q : ℕ, Nat.Prime p ∧ Nat.Prime q ∧ r < p ∧ r < q ∧ p ≠ q := by | |
| obtain ⟨p, hrp, hp⟩ := Nat.exists_infinite_primes (r + 1) | |
| obtain ⟨q, hpq, hq⟩ := Nat.exists_infinite_primes (p + 1) | |
| exact ⟨p, q, hp, hq, by omega, by omega, by omega⟩ | |
| end IntegralCharacterVarieties.Arithmetic | |
| namespace IntegralCharacterVarieties | |
| theorem density_transfer | |
| {ι Y : Type*} [TopologicalSpace Y] | |
| (B T : ι → Type*) [∀ i, TopologicalSpace (B i)] | |
| [∀ i, TopologicalSpace (T i)] | |
| (p : ∀ i, T i → B i) (f : ∀ i, T i → Y) | |
| (hp : ∀ i, IsOpenMap (p i)) (hf : ∀ i, Continuous (f i)) | |
| (hcover : ∀ y : Y, ∃ (i : ι) (t : T i), f i t = y) | |
| (I : ∀ i, Set (B i)) (hI : ∀ i, Dense (I i)) | |
| (J : Set Y) | |
| (hfiber : ∀ (i : ι) (b : B i), b ∈ I i → | |
| Dense {t : {t : T i // p i t = b} | f i t.1 ∈ J}) : | |
| Dense J := by | |
| rw [dense_iff_inter_open] | |
| intro U hU hUnon | |
| obtain ⟨y, hy⟩ := hUnon | |
| obtain ⟨i, t, hty⟩ := hcover y | |
| have hV : IsOpen (f i ⁻¹' U) := (hf i).isOpen_preimage U hU | |
| have htV : t ∈ f i ⁻¹' U := by | |
| change f i t ∈ U | |
| rw [hty] | |
| exact hy | |
| have hnon : (p i '' (f i ⁻¹' U)).Nonempty := ⟨p i t, t, htV, rfl⟩ | |
| obtain ⟨b, hbI, t', ht'V, ht'b⟩ := | |
| (hI i).exists_mem_open (hp i _ hV) hnon | |
| let t₀ : {t : T i // p i t = b} := ⟨t', ht'b⟩ | |
| have hW : IsOpen {t : {t : T i // p i t = b} | f i t.1 ∈ U} := | |
| ((hf i).comp continuous_subtype_val).isOpen_preimage U hU | |
| have hWnon : {t : {t : T i // p i t = b} | f i t.1 ∈ U}.Nonempty := | |
| ⟨t₀, ht'V⟩ | |
| obtain ⟨s, hsJ, hsU⟩ := (hfiber i b hbI).exists_mem_open hW hWnon | |
| exact ⟨f i s.1, hsU, hsJ⟩ | |
| end IntegralCharacterVarieties | |
| end | |
| end OAI | |