integer — Mathlib

English

Define the R-subalgebra of the S-integers of K as the intersection of valuation subrings over all places not in S.

Русский

Определяем R-подалгебру S-интеграв в K как пересечение valuation-обменных колец над всеми местами вне S.

LaTeX

$$$$\\text{integer} : \\text{Subalgebra } R K$$$$

Lean4

/-- The `R`-subalgebra of `S`-integers of `K`. -/
@[simps!]
def integer : Subalgebra R K :=
  {
    (⨅ (v) (_ : v ∉ S), (v.valuation K).valuationSubring.toSubring).copy
        {x : K | ∀ (v) (_ : v ∉ S), v.valuation K x ≤ 1} <|
      Set.ext fun _ => by simp [SetLike.mem_coe] with
    algebraMap_mem' := fun x v _ => v.valuation_le_one x }

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