Theorems · Theorem · field theory

sum_smul_minpolyDiv_eq_X_pow

∀ {K : Type u_2} {L : Type u_3} [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (E : Type u_1)
  [inst_3 : Field E] [inst_4 : Algebra K E] [IsAlgClosed E] [inst_6 : FiniteDimensional K L] [Algebra.IsSeparable K L]
  {x : L},
  K[x] = ⊤ →
    ∀ {r : ℕ},
      r < Module.finrank K L →
        ∑ σ,
            Polynomial.map (↑σ)
              ((x ^ r / (Polynomial.aeval x) (Polynomial.derivative (minpoly K x))) • minpolyDiv K x) =
          Polynomial.X ^ r

Defined in

Mathlib.FieldTheory.Minpoly.MinpolyDiv

Cited by

1 results in Mathlib

Foundations

Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraFieldAlgebraIsAlgClosedFiniteDimensionalAlgebra.IsSeparable