isDiscreteValuationRing — Mathlib

English

In the relation between Verschiebung and Frobenius maps on Witt vectors, any nonzero element can be written as p^m times another Witt vector with a unit coefficient.

Русский

В отношении Verschiebung и Frobenius на Witt-векторах каждое ненулевое элемент может быть записано как p^m на другой Witt-вектор с единичным коэффициентом.

LaTeX

$$$\exists m,b: a = p^m \cdot b \quad (b_0 \neq 0)$$$

Lean4

/-- The ring of Witt Vectors of a perfect field of positive characteristic is a DVR.
-/
instance isDiscreteValuationRing : IsDiscreteValuationRing (𝕎 k) :=
  IsDiscreteValuationRing.ofHasUnitMulPowIrreducibleFactorization
    (by
      refine ⟨p, irreducible p, fun {x} hx => ?_⟩
      obtain ⟨n, b, hb⟩ := exists_eq_pow_p_mul' x hx
      exact ⟨n, b, hb.symm⟩)

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