uniqueDifferentialAlgebraFiniteDimensional

English

For irreducible monic p over F, AdjoinRoot p is equipped with a differential-algebra structure; and this structure is compatible with the integral embedding.

Русский

Для ирреducible монического p над F, AdjoinRoot p имеет структуру дифференциальной алгебры, которая совместима с интегральной вложенностью.

LaTeX

$$$\\text{DifferentialAlgebra } F (\\operatorname{AdjoinRoot}(p))$$$

Lean4

/-- A finite extension of a differential field has a unique derivation which agrees with the one on the
base field.
-/
noncomputable def uniqueDifferentialAlgebraFiniteDimensional [FiniteDimensional F K] :
    Unique { _a : Differential K // DifferentialAlgebra F K } :=
  by
  let default : { _a : Differential K // DifferentialAlgebra F K } :=
    ⟨differentialFiniteDimensional F K, differentialAlgebraFiniteDimensional⟩
  refine ⟨⟨default⟩, fun ⟨a, ha⟩ ↦ ?_⟩
  ext x
  apply_fun (aeval x (mapCoeffs (minpoly F x)) + aeval x (derivative (minpoly F x)) * ·)
  · conv_lhs => apply (deriv_aeval_eq ..).symm
    conv_rhs => apply (@deriv_aeval_eq _ _ _ _ _ default.1 _ default.2 _ _).symm
    simp
  · apply (add_right_injective _).comp
    apply mul_right_injective₀
    rw [ne_eq, ← minpoly.dvd_iff]
    have : 0 < (minpoly F x).natDegree :=
      Irreducible.natDegree_pos (minpoly.irreducible (Algebra.IsIntegral.isIntegral _))
    apply not_dvd_of_natDegree_lt
    · intro nh
      simp [natDegree_eq_zero_of_derivative_eq_zero nh] at this
    apply natDegree_derivative_lt
    exact Nat.ne_zero_of_lt this

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