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feat(FieldTheory/IntermediateField): adjoining a shifted or scaled generator #41649

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feat(FieldTheory/IntermediateField): adjoining a shifted or scaled generator #41649

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feat(FieldTheory/IntermediateField): adjoining a shifted or scaled ge…
xroblot Jul 12, 2026
24085d1
chore: rename to adjoin_simple_{add,mul}_algebraMap
xroblot Jul 12, 2026
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21 changes: 21 additions & 0 deletions Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -254,6 +254,27 @@ theorem exists_finset_of_mem_adjoin {S : Set E} {x : E} (hx : x ∈ adjoin F S)
rintro _ h _ rfl
exact subset_adjoin F _ ⟨_, h, rfl⟩

/-- Adjoining `x + algebraMap F E y`, where `y` lies in the base field, yields the same simple
extension as adjoining `x`. -/
theorem adjoin_simple_add_algebraMap (x : E) (y : F) : F⟮x + algebraMap F E y⟯ = F⟮x⟯ := by
apply le_antisymm
· rw [adjoin_le_iff, Set.singleton_subset_iff, SetLike.mem_coe]
exact add_mem (mem_adjoin_simple_self F x) (algebraMap_mem _ y)
· rw [adjoin_simple_le_iff]
convert IntermediateField.sub_mem _ (mem_adjoin_simple_self F _) (algebraMap_mem _ y)
rw [eq_sub_iff_add_eq]

/-- Adjoining `x * algebraMap F E y`, where `y` is a nonzero element of the base field, yields the
same simple extension as adjoining `x`. -/
theorem adjoin_simple_mul_algebraMap (x : E) (y : F) (hy : y ≠ 0) :
F⟮x * algebraMap F E y⟯ = F⟮x⟯ := by
apply le_antisymm
· rw [adjoin_le_iff, Set.singleton_subset_iff, SetLike.mem_coe]
exact mul_mem (mem_adjoin_simple_self F x) (algebraMap_mem _ y)
· rw [adjoin_simple_le_iff]
convert IntermediateField.div_mem _ (mem_adjoin_simple_self F _) (algebraMap_mem _ y)
rw [mul_div_cancel_right₀ x (by rwa [_root_.map_ne_zero])]

end AdjoinDef

section AdjoinIntermediateFieldLattice
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