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 \({{\rm cd}_2(F) \leq 2}\) 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 \({{\rm cd}_2(F) \leq 2}\) as a tensor product of quaternion algebras.
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Barry, D., Chapman, A. Square-central and Artin–Schreier elements in division algebras. Arch. Math. 104, 513–521 (2015). https://doi.org/10.1007/s00013-015-0773-2
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DOI: https://doi.org/10.1007/s00013-015-0773-2