minpoly_squarefree — Mathlib

English

For a semisimple endomorphism f, its minimal polynomial is square-free.

Русский

Для семисемпсимого f минимальный полином квадратно-свободен.

LaTeX

$$$ IsSemisimple(f) \Rightarrow Squarefree(minpoly_K(f))$$$

Lean4

/-- The minimal polynomial of a semisimple endomorphism is square free -/
theorem minpoly_squarefree : Squarefree (minpoly K f) :=
  IsRadical.squarefree (minpoly.ne_zero <| Algebra.IsIntegral.isIntegral _) <| by
    rw [isRadical_iff_span_singleton, span_minpoly_eq_annihilator]; exact hf.annihilator_isRadical

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