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e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | 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
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