| import OAI.RingTheory.DirectFiniteness.FrameExistence | |
| namespace OAI | |
| noncomputable section | |
| open Module | |
| namespace KaplanskyCounterexample.FiniteGeometry | |
| /-- Finite characteristic-two fields have finite extensions of every positive degree. -/ | |
| theorem exists_finite_extension (Q : Type) [Field Q] [Fintype Q] [CharP Q 2] | |
| (b : ℕ) (hb : 0 < b) : | |
| ∃ (F : Type) (_ : Field F) (_ : Algebra Q F) (_ : Fintype F), | |
| finrank Q F = b := by | |
| let : Algebra (ZMod 2) Q := ZMod.algebra Q 2 | |
| let h := finrank (ZMod 2) Q | |
| have hh : 0 < h := Module.finrank_pos | |
| let F := GaloisField 2 (h * b) | |
| have hFb : finrank (ZMod 2) F = h * b := | |
| GaloisField.finrank 2 (Nat.ne_of_gt (Nat.mul_pos hh hb)) | |
| obtain ⟨f⟩ := FiniteField.nonempty_algHom_of_finrank_dvd (K := Q) (L := F) | |
| (by rw [hFb]; exact dvd_mul_right h b) | |
| algebraize [f.toRingHom] | |
| let : Fintype F := Fintype.ofFinite F | |
| refine ⟨F, inferInstance, inferInstance, inferInstance, ?_⟩ | |
| have ht := Module.finrank_mul_finrank (ZMod 2) Q F | |
| rw [hFb] at ht | |
| exact Nat.eq_of_mul_eq_mul_left hh ht | |
| end KaplanskyCounterexample.FiniteGeometry | |
| end | |
| end OAI | |