English
For prime P and Q over p, the ramification indices with respect to the base map algebraMap A B are equal: ramificationIdx (algebraMap A B) p P = ramificationIdx (algebraMap A B) p Q.
Русский
Для простых P и Q над p верна равенство ромификационных индексов: ramificationIdx (algebraMap A B) p P = ramificationIdx (algebraMap A B) p Q.
LaTeX
$$$\\mathrm{ramificationIdx}(\\mathrm{algebraMap}\\ A\\ B)\\ p\\ P = \\mathrm{ramificationIdx}(\\mathrm{algebraMap}\\ A\\ B)\\ p\\ Q$$$
Lean4
theorem isPretransitive_of_isGalois [IsGalois K L] : MulAction.IsPretransitive (B ≃ₐ[A] B) (primesOver p B) where
exists_smul_eq := by
intro ⟨P, _, _⟩ ⟨Q, _, _⟩
rcases exists_map_eq_of_isGalois p P Q K L with ⟨σ, hs⟩
exact ⟨σ, Subtype.val_inj.mp hs⟩