isUnit_leadingCoeff_of_dvd — Mathlib

English

For a monic p ∈ R[x] with p ≠ 1, irreducibility of p is equivalent to the property that every divisor q with degree ≤ deg(p)/2 is a unit.

Русский

Для моничного p ∈ R[x] с p ≠ 1, неприводимость эквивалентна свойству: любой делитель q с deg(q) ≤ deg(p)/2 — единица.

LaTeX

$$Monic p ∧ p ≠ 1 → (Irreducible p ↔ ∀ q, deg q ≤ deg(p)/2 → q | p → IsUnit q)$$

Lean4

theorem isUnit_leadingCoeff_of_dvd {a p : R[X]} (hp : Monic p) (hap : a ∣ p) : IsUnit a.leadingCoeff :=
  isUnit_of_dvd_one (by simpa only [hp.leadingCoeff] using leadingCoeff_dvd_leadingCoeff hap)

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