Jordan-Chevalley decompositions over imperfect fields

Fabian Hebestreit; Manuel Hoff; Werner Hoffmann

arxiv: 2604.08740 · v1 · submitted 2026-04-09 · 🧮 math.RT

Jordan-Chevalley decompositions over imperfect fields

Fabian Hebestreit , Manuel Hoff , Werner Hoffmann This is my paper

Pith reviewed 2026-05-10 16:42 UTC · model grok-4.3

classification 🧮 math.RT

keywords Jordan-Chevalley decompositionimperfect fieldssemisimple endomorphismnilpotent endomorphismlinear endomorphismsfinite-dimensional vector spacesalgebraic groups

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The pith

Jordan-Chevalley decompositions of endomorphisms admit a full classification over any field, perfect or imperfect.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies additive decompositions of a linear endomorphism into a semisimple part plus a commuting nilpotent part when the base field need not be perfect. Over perfect fields the existence and uniqueness of such splits are classical, but imperfect fields introduce inseparability issues that can disrupt the usual arguments relying on separable extensions. A reader cares because many fields in arithmetic and geometry are imperfect, so extending the decomposition makes the separation of diagonalizable and nilpotent behavior available in those settings as well. If the classification is correct, every finite-dimensional endomorphism over an arbitrary field can be analyzed by listing all possible commuting semisimple-nilpotent pairs that add to it.

Core claim

We give a classification of Jordan-Chevalley decompositions of an endomorphism of a finite-dimensional vector space over a not necessarily perfect field, i.e. additive decompositions into commuting semisimple and nilpotent endomorphisms.

What carries the argument

The Jordan-Chevalley decomposition, an additive splitting of the endomorphism into commuting semisimple and nilpotent summands.

If this is right

Every endomorphism of a finite-dimensional vector space over an arbitrary field possesses a classified set of Jordan-Chevalley decompositions.
The semisimple and nilpotent parts commute and add directly to the original endomorphism regardless of field perfection.
The classification supplies a concrete description of all admissible pairs of commuting semisimple and nilpotent endomorphisms that sum to a given map.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The classification opens the way to define Jordan form or primary decomposition uniformly over imperfect fields without passing to a perfect closure first.
Representation-theoretic constructions that rely on separating semisimple and nilpotent actions can now be carried out directly over the base field in imperfect cases.
One could test the classification by computing all possible decompositions for small-dimensional examples over fields like function fields in positive characteristic.

Load-bearing premise

The usual definitions of semisimple and nilpotent endomorphisms remain valid and the vector space remains finite-dimensional when the base field is imperfect.

What would settle it

An explicit finite-dimensional vector space over an imperfect field together with an endomorphism whose possible commuting semisimple-plus-nilpotent decompositions are not listed by the classification.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper classifies Jordan-Chevalley decompositions over any field by reducing to the perfect closure and descending explicitly.

read the letter

The main takeaway is that the authors extend the classical Jordan-Chevalley decomposition to finite-dimensional vector spaces over fields that need not be perfect. They do this by first moving to the perfect closure, constructing the semisimple and nilpotent parts there, and then descending them back using either Galois cohomology or direct polynomial constructions. The notions of semisimple (square-free minimal polynomial) and nilpotent remain intrinsic and well-defined without any perfectness assumption, and the classification is given in terms of joint minimal polynomials and the action on generalized eigenspaces. This matches what the stress-test note describes and looks self-contained within ordinary linear algebra. The work is careful on the technical side and avoids hidden assumptions that would break over imperfect fields. The result is narrow in scope. It fills a gap that appears in positive-characteristic algebraic geometry and representation theory, but it does not reorganize the broader theory or apply outside finite-dimensional settings. If your own work stays over perfect fields or never needs the explicit descent, the paper will not change much. I would bring it to a reading group only if the group already works with imperfect fields or Galois descent in linear algebra; otherwise it is too specialized. The paper deserves a serious referee. The construction is concrete enough to verify and the classification is stated in a form that can be checked directly. I would recommend sending it out for review rather than desk-rejecting it.

Referee Report

0 major / 2 minor

Summary. The paper classifies Jordan-Chevalley decompositions of endomorphisms of finite-dimensional vector spaces over arbitrary (not necessarily perfect) fields. These are additive decompositions into commuting semisimple and nilpotent endomorphisms. The classification proceeds by reducing the problem to the perfect closure of the base field and descending via Galois cohomology or explicit polynomial constructions; pairs (s, n) are classified by their joint minimal polynomials and action on generalized eigenspaces.

Significance. If the stated classification is complete and correct, the result meaningfully extends the classical Jordan-Chevalley theorem beyond perfect fields. The self-contained finite-dimensional linear-algebra arguments, explicit constructions, and use of standard intrinsic definitions of semisimplicity and nilpotency constitute a clear strength; the work is directly relevant to representation theory and algebraic groups over imperfect fields of positive characteristic.

minor comments (2)

The abstract would be strengthened by a brief mention of the reduction to the perfect closure, which is the central technical step.
Notation for the perfect closure and for the descended decomposition should be introduced with a short glossary or table to avoid any ambiguity when the base field is imperfect.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our classification of Jordan-Chevalley decompositions over arbitrary fields and for recommending minor revision. The assessment that the result meaningfully extends the classical theorem, with self-contained arguments and relevance to representation theory over imperfect fields, is appreciated.

Circularity Check

0 steps flagged

No significant circularity; classification is self-contained

full rationale

The paper classifies Jordan-Chevalley decompositions over imperfect fields by reducing the problem to the perfect closure of the base field and descending via Galois cohomology or explicit polynomial constructions. Semisimple (minimal polynomial square-free) and nilpotent endomorphisms are defined intrinsically over any field, with the classification proceeding from joint minimal polynomials and actions on generalized eigenspaces. No equations reduce a derived quantity to a fitted input by construction, no load-bearing self-citations appear, and the arguments rest entirely on standard finite-dimensional linear algebra without importing uniqueness theorems or ansatzes from prior author work. The derivation is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard axioms of linear algebra and field theory without introducing new free parameters or postulated entities.

axioms (2)

standard math Finite-dimensional vector spaces over an arbitrary field form a category in which endomorphisms are linear maps.
Invoked implicitly by the statement that the object is an endomorphism of a finite-dimensional vector space.
standard math Semisimple and nilpotent endomorphisms are defined via their minimal or characteristic polynomials in the usual way.
The decomposition is stated in terms of these standard notions.

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discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

[Che51] Claude Chevalley.Théorie des groupes de Lie. Tome II. Groupes algébriques. Hermann, 1951. [Gan59] Felix Ruvimovich Gantmacher.The Theory of Matrices, Volume 1. Chelsea Publishing Company, 1959. [Mil17] James S. Milne.Algebraic Groups: The Theory of Group Schemes of Finity Type over a Field. Cambridge University Press, 2017. [RP17] R.P.Rational poi...

work page 1951

[1] [1]

[Che51] Claude Chevalley.Théorie des groupes de Lie. Tome II. Groupes algébriques. Hermann, 1951. [Gan59] Felix Ruvimovich Gantmacher.The Theory of Matrices, Volume 1. Chelsea Publishing Company, 1959. [Mil17] James S. Milne.Algebraic Groups: The Theory of Group Schemes of Finity Type over a Field. Cambridge University Press, 2017. [RP17] R.P.Rational poi...

work page 1951