map_ne_one — Mathlib

English

If p is monic, p(x) = 0, deg p = n, and the set {x^i} for i < n is linearly independent over A, then p = minpoly_A(x).

Русский

Если p моноический, p(x)=0, deg p = n, и множество {x^i} (i < n) линейно независимо над A, то p = minpoly_A(x).

LaTeX

$$$p\\text{ мон..ий } p(x)=0,\\ deg p = n,\\text{ и } \\{x^i\\}_{i

Lean4

theorem map_ne_one [Nontrivial B] {R : Type*} [Semiring R] [Nontrivial R] (f : A →+* R) : (minpoly A x).map f ≠ 1 :=
  by
  by_cases hx : IsIntegral A x
  · exact mt ((monic hx).eq_one_of_map_eq_one f) (ne_one A x)
  · rw [eq_zero hx, Polynomial.map_zero]
    exact zero_ne_one

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