Theorems · Theorem · field theory

Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors

∀ {R : Type u} [inst : CommRing R] {p : Polynomial R} {t : R} {n : ℕ},
  p ≠ 0 →
    ↑n.factorial ∈ nonZeroDivisors R →
      (n < Polynomial.rootMultiplicity t p ↔ ∀ m ≤ n, ((⇑Polynomial.derivative)^[m] p).IsRoot t)

Defined in: Mathlib.Algebra.Polynomial.FieldDivision
Cited by: 1 results in Mathlib
Foundations: Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing

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Cites15

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
RingHom.idstatement · cited by 18,349
CommRingstatement and proof · cited by 17,173
LinearMapstatement · cited by 10,215
Polynomialstatement and proof · cited by 5,681
Submonoidstatement · cited by 3,086
nonZeroDivisorsstatement and proof · cited by 895
LE.le.trans_ltproof · cited by 795
Nat.iteratestatement and proof · cited by 740
Nat.factorialstatement and proof · cited by 616
Polynomial.derivativestatement and proof · cited by 331
Polynomial.IsRootstatement and proof · cited by 152

Cited by1

Results whose statement or proof uses this declaration.

Polynomial.one_lt_rootMultiplicity_iff_isRoot_iterate_derivativeproof · cited by 1