| import OAI.RingTheory.DirectFiniteness.PrescribedDirections |
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| namespace OAI |
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| universe u_1 u_2 u_3 |
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| noncomputable section |
| open scoped Classical BigOperators |
| namespace KaplanskyCounterexample.CoordinateCharacters |
| variable (ℓ : ℕ) [Fact ℓ.Prime] |
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| local instance modulusNeZero : NeZero ℓ := inferInstance |
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|
| abbrev Plane := Multiplicative (Fin 2 → ZMod ℓ) |
| abbrev Cyclic := Multiplicative (ZMod ℓ) |
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| def form (a : Fin 2 → ZMod ℓ) : Plane ℓ →* Cyclic ℓ := |
| AddMonoidHom.toMultiplicative |
| ({ toFun := fun x => ∑ i : Fin 2, a i * x i |
| map_zero' := by simp |
| map_add' := by intro x y; simp [mul_add, Finset.sum_add_distrib] } : |
| (Fin 2 → ZMod ℓ) →+ ZMod ℓ) |
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| lemma form_injective : Function.Injective (form ℓ) := by |
| intro a b hab |
| funext i |
| have hi := congrArg (fun φ : Plane ℓ →* Cyclic ℓ => |
| Multiplicative.toAdd (φ (Multiplicative.ofAdd (Pi.single i 1)))) hab |
| fin_cases i <;> simpa [form, Pi.single_apply] using hi |
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| lemma exists_slot_forms {V : Type u_1} {m : Type u_2} [Fintype V] [Fintype m] |
| (hcard : Fintype.card V ≤ Fintype.card m * ℓ ^ 2) : |
| ∃ (slot : V → m) (α : V → Plane ℓ →* Cyclic ℓ), |
| ∀ v w, v ≠ w → slot v = slot w → α v ≠ α w := by |
| have hc : Fintype.card V ≤ Fintype.card (m × (Fin 2 → ZMod ℓ)) := by |
| simpa using hcard |
| obtain ⟨f⟩ : Nonempty (V ↪ m × (Fin 2 → ZMod ℓ)) := Function.Embedding.nonempty_of_card_le hc |
| refine ⟨fun v => (f v).1, fun v => form ℓ (f v).2, ?_⟩ |
| intro v w hvw hs ha |
| exact hvw (f.injective (Prod.ext hs (form_injective ℓ ha))) |
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| variable {K : Type u_3} [Field K] |
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| def cyclicCharacter {ζ : K} (hζ : IsPrimitiveRoot ζ ℓ) : Cyclic ℓ →* Kˣ := |
| (AddChar.zmodChar ℓ hζ.pow_eq_one).toMonoidHom.toHomUnits |
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| lemma cyclicCharacter_injective {ζ : K} (hζ : IsPrimitiveRoot ζ ℓ) : |
| Function.Injective (cyclicCharacter ℓ hζ) := by |
| intro a b hab |
| apply Multiplicative.toAdd.injective |
| apply ZMod.val_injective ℓ |
| apply hζ.pow_inj (ZMod.val_lt _) (ZMod.val_lt _) |
| exact congrArg (fun u : Kˣ => (u : K)) hab |
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| lemma cyclicCharacter_ne_one {ζ : K} (hζ : IsPrimitiveRoot ζ ℓ) : |
| cyclicCharacter ℓ hζ ≠ 1 := by |
| intro h |
| have he : Multiplicative.ofAdd (1 : ZMod ℓ) = 1 := |
| (cyclicCharacter_injective ℓ hζ) (by simp [h]) |
| exact (one_ne_zero : (1 : ZMod ℓ) ≠ 0) (congrArg Multiplicative.toAdd he) |
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| end KaplanskyCounterexample.CoordinateCharacters |
| end |
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| end OAI |
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