isPretransitive_of_isGalois — Mathlib

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⟩

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