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.

18 pages