A Reduced-Trace-Zero Element That Is Not a Commutator in a Central Division Algebra

Abstract

We exhibit a finite-dimensional central division algebra, of degree four over its center, together with an explicit element of reduced trace zero that is not a single commutator. This answers negatively the single-commutator question for finite-dimensional central division algebras, the case left open by Amitsur and Rowen. The algebra is a skew Laurent series ring, the element has only two terms, and the proof uses the $x$-adic valuation, a first-obstruction lemma based on two commuting involutions of the associated graded ring, and a parity argument for quadratic forms over iterated Laurent series fields.

Disclosure

“Section 2 and, from them, conjectured the existence of counterexamples. ChatGPT-5 was subse- quently used to assist in the search for explicit candidate counterexamples. The reduction argument in Section 4 was developed by the author, with ChatGPT-5 used to assist with calculations and to check intermediate steps. During the prepa- ration of the final manuscript, ChatGPT-5 was also used for language polishing and grammatical correction. Claude Opus was used only for independent check”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T19:01:43.215179Z

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