ramificationIdx_eq_factors_count — Mathlib

English

In a Dedekind domain S, the ramification index e over p at a prime P equals the number of times P occurs in the factorization of the image of p under f.

Русский

В доказовом домене S величина рамфикации e над p при простом P равна количеству повторений P в разложении образа p по f.

LaTeX

$$$\\\\operatorname{ramificationIdx} f p P = (\\\\text{factors} (\\\\operatorname{map} f p)).\\\\count P$$$

Lean4

theorem ramificationIdx_eq_factors_count [DecidableEq (Ideal S)] (hp0 : map f p ≠ ⊥) (hP : P.IsPrime) (hP0 : P ≠ ⊥) :
    ramificationIdx f p P = (factors (map f p)).count P := by
  rw [IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hp0 hP hP0, factors_eq_normalizedFactors]

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