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[Merged by Bors] - refactor(NumberTheory): golf Mathlib/NumberTheory/Cyclotomic/Gal #38495

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yuanyi-350 wants to merge 2 commits into from
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[Merged by Bors] - refactor(NumberTheory): golf Mathlib/NumberTheory/Cyclotomic/Gal #38495

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refactor(NumberTheory): golf Cyclotomic/Gal
yuanyi-350 Apr 25, 2026
a0b8d51
avoid defeq abuse
yuanyi-350 Apr 26, 2026
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21 changes: 4 additions & 17 deletions Mathlib/NumberTheory/Cyclotomic/Gal.lean
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Expand Up @@ -56,24 +56,11 @@ variable [CommRing L] [IsDomain L] (hμ : IsPrimitiveRoot μ n) [Algebra K L]
field extension. -/
theorem autToPow_injective : Function.Injective <| hμ.autToPow K := by
intro f g hfg
apply_fun Units.val at hfg
simp only [IsPrimitiveRoot.coe_autToPow_apply] at hfg
generalize_proofs hf' hg' at hfg
have hf := hf'.choose_spec
have hg := hg'.choose_spec
generalize_proofs hζ at hf hg
suffices f (hμ.toRootsOfUnity : Lˣ) = g (hμ.toRootsOfUnity : Lˣ) by
apply AlgEquiv.coe_algHom_injective
have : f.toAlgHom = g.toAlgHom := by
apply (hμ.powerBasis K).algHom_ext
exact this
rw [ZMod.natCast_eq_natCast_iff] at hfg
refine (hf.trans ?_).trans hg.symm
rw [← rootsOfUnity.coe_pow _ hf'.choose, ← rootsOfUnity.coe_pow _ hg'.choose]
congr 2
rw [pow_eq_pow_iff_modEq]
convert hfg
conv => enter [2]; rw [hμ.eq_orderOf, ← hμ.val_toRootsOfUnity_coe]
rw [orderOf_units, Subgroup.orderOf_coe]
rw [AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe, AlgHom.coe_coe, AlgHom.coe_coe,
powerBasis_gen, ← autToPow_spec K hμ g, ← autToPow_spec K hμ f, hfg]
exact AlgEquiv.coe_algHom_injective this

end IsPrimitiveRoot

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