Compatible derivations commute with field trace

Proved

OAI.PiExponent.derivation_trace_of_basis

by marwahaha · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

differential-algebranumber-theoryopenai-pi-exponent-017

Let $k\subseteq F\subseteq E$ be a compatible field tower, and assume a finite $F$ -basis of $E$ is given. If $k$ -derivations $d$ on $F$ and $D$ on $E$ agree on the embedded copy of $F$ , then every $x\in E$ satisfies

d(\operatorname{Tr}_{E/F}(x))=\operatorname{Tr}_{E/F}(D(x)).

This relates differentiation in a finite extension to differentiation in its base field.

Preamble

import Mathlib.Algebra.Polynomial.Derivation
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.LaurentSeries
import Mathlib.RingTheory.Trace.Defs
import Mathlib.Tactic


namespace OAI
namespace PiExponent
end PiExponent
end OAI
open scoped LaurentSeries Matrix
open Polynomial
open OAI.PiExponent

Formal statement

theorem OAI.PiExponent.derivation_trace_of_basis
    {k F E ι : Type*} [Field k] [Field F] [Field E] [Fintype ι]
    [Algebra k F] [Algebra k E] [Algebra F E] [IsScalarTower k F E]
    (d : Derivation k F F) (D : Derivation k E E)
    (hcompat : ∀ a : F, D (algebraMap F E a) = algebraMap F E (d a))
    (b : Module.Basis ι F E) (x : E) :
    d (Algebra.trace F E x) = Algebra.trace F E (D x) := by sorry

Source

https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/NumberTheory/PiExponent/Analysis/LogarithmicObstruction.lean#L126-L158

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