Square-Central and Artin-Schreier Elements in Division Algebras

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 ${\mathrm{cd}}_2(F)\le 2$, 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 ${\mathrm{cd}}_2(F)\le 2$ as a tensor product of quaternion algebras.