Autour de la d\'ecomposition des alg\`ebres d'exposant 2 sur les extensions multiquadratiques

TL;DR

This paper investigates the decomposition of exponent 2 central simple algebras over fields of characteristic 2 with 2-cohomological dimension 2, extending properties to multiquadratic extensions and providing new examples and applications.

Contribution

It extends known decomposition properties to multiquadratic extensions of separability degree up to 4 and constructs a specific algebra illustrating limitations of adapted decompositions in characteristic 2.

Findings

Extended properties to multiquadratic extensions of separability degree ≤ 4

Constructed an algebra of exponent 2 and degree 8 with specific extension properties

Provided an elementary proof of non-excellence of separable biquadratic extensions

Abstract

For central simple algebras of exponent $2$ over fields of characteristic $2$ and $2$-cohomological dimension equal to $2$, we study the adapted decomposition to some multiquadratic extensions of the base field. Several remarkable properties are extended to multiquadratic extensions of separability degree at most $4$. We also extend to the characteristic $2$ a result of Elman-Lam-Tignol-Wadsworth by constructing an algebra of exponent $2$ and degree $8$ containing a separable triquadratic extension but which admits no adapted decomposition to this extension. As an application we give an elementary proof of the non-excellence of separable biquadratic extensions.