Square-Central and Artin-Schreier Elements in Division Algebras

TL;DR

This paper investigates the properties of special elements in division algebras of exponent 2 and degree a power of 2, providing new lemmas and extending known theorems to characteristic 2 fields.

Contribution

It introduces chain lemmas for square-central and Artin-Schreier elements in division algebras over 2-fields with low cohomological dimension, and extends Merkurjev's theorem to characteristic 2.

Findings

Established chain lemmas for specific elements in division algebras.

Deduced a common slot lemma for tensor products of quaternion algebras.

Extended Merkurjev's theorem to characteristic 2 fields.

Abstract

We study the behavior of square-central elements and Artin-Schreier elements in division algebras of exponent 2 and degree a power of 2. We provide chain lemmas for such elements in division algebras over 2-fields $F$ of cohomological $2$-dimension $\operatorname{cd}_2(F) \leq 2$, and deduce a common slot lemma for tensor products of quaternion algebras over such fields. We also extend to characteristic 2 a theorem proven by Merkurjev for characteristic not 2 on the decomposition of any central simple algebra of exponent 2 and degree a power of 2 over a field $F$ with $\operatorname{cd}_2(F) \leq 2$ as a tensor product of quaternion algebras.