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perf(FieldTheory/PurelyInseparable): golf proof #41664

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25 changes: 13 additions & 12 deletions Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean
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Expand Up @@ -234,20 +234,21 @@ instance isPurelyInseparable_iSup {ι : Sort*} {t : ι → IntermediateField F E
theorem adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable (S : Set E)
[Algebra.IsSeparable F (adjoin F S)] (q : ℕ) [ExpChar F q] (n : ℕ) :
adjoin F S = adjoin F ((· ^ q ^ n) '' S) := by
set L := adjoin F S
set M := adjoin F ((· ^ q ^ n) '' S)
have hi : M ≤ L := by
rw [adjoin_le_iff]
rintro _ ⟨y, hy, rfl⟩
have := expChar_of_injective_algebraMap (algebraMap F M).injective q
refine le_antisymm (adjoin_le_iff.2 fun x hx ↦ ?_) (adjoin_le_iff.2 ?_)
· have : Algebra.IsSeparable M (adjoin M {x}) :=

@acmepjz acmepjz Jul 13, 2026

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Suggested change
· have : Algebra.IsSeparable M (adjoin M {x}) :=
· have : Algebra.IsSeparable M M⟮x⟯ :=

Maybe this looks better (untested) Same as two adjoin M {x} below.

(isSeparable_adjoin_simple_iff_isSeparable M E).2 <|
((isSeparable_adjoin_iff_isSeparable F E).1 inferInstance x hx).tower_top M
have : IsPurelyInseparable M (adjoin M {x}) :=
(isPurelyInseparable_adjoin_simple_iff_pow_mem M E q).2
⟨n, ⟨x ^ q ^ n, subset_adjoin F _ ⟨x, hx, rfl⟩⟩, rfl⟩
have hx' := mem_adjoin_simple_self M x
rw [(adjoin M {x}).eq_bot_of_isPurelyInseparable_of_isSeparable, mem_bot] at hx'
obtain ⟨y, rfl⟩ := hx'
exact y.2
· rintro _ ⟨y, hy, rfl⟩
exact pow_mem (subset_adjoin F S hy) _
letI := (inclusion hi).toAlgebra
haveI : Algebra.IsSeparable M (extendScalars hi) :=
Algebra.isSeparable_tower_top_of_isSeparable F M L
haveI : IsPurelyInseparable M (extendScalars hi) := by
rw [extendScalars_adjoin hi, isPurelyInseparable_adjoin_iff_pow_mem M _ q]
exact fun x hx ↦ ⟨n, ⟨x ^ q ^ n, subset_adjoin F _ ⟨x, hx, rfl⟩⟩, rfl⟩
simpa only [extendScalars_restrictScalars, restrictScalars_bot_eq_self] using congr_arg
(restrictScalars F) (extendScalars hi).eq_bot_of_isPurelyInseparable_of_isSeparable

/-- If `E / F` is a separable field extension of exponential characteristic `q`, then
`F(S) = F(S ^ (q ^ n))` for any subset `S` of `E` and any natural number `n`. -/
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