of_irreducible_expand_pow — Mathlib

English

If p ≠ 0 and expand_R(p^n,f) is irreducible, then f is irreducible.

Русский

Если p ≠ 0 и expand_R(p^n,f) иррефуберен, то f иррефуберен.

LaTeX

$$$$\text{Irreducible}(\mathrm{expand}_R(p^n,f))\Rightarrow \text{Irreducible}(f)$$$$

Lean4

theorem of_irreducible_expand_pow {p : ℕ} (hp : p ≠ 0) {f : R[X]} {n : ℕ} :
    Irreducible (expand R (p ^ n) f) → Irreducible f :=
  Nat.recOn n (fun hf => by rwa [pow_zero, expand_one] at hf) fun n ih hf =>
    ih <|
      of_irreducible_expand hp <| by
        rw [pow_succ'] at hf
        rwa [expand_expand]

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