A cohomological invariant for algebras of degree 8 and exponent 2 in characteristic 2

TL;DR

This paper introduces a new cohomological invariant for certain degree 8 algebras in characteristic 2, extending previous work and linking algebra decomposability to invariant vanishing.

Contribution

It extends Sivatski's work to characteristic 2 by defining a cohomological invariant for degree 8 algebras and explores its applications in algebra decomposability and descent problems.

Findings

The invariant $ ext{inv}(A)$ is well-defined for specific algebras in characteristic 2.

Decomposability of degree 8 algebras relates to the vanishing of $ ext{inv}(A)$.

The invariant aids in descent results for algebras and quadratic forms over biquadratic extensions.

Abstract

Our aim in this paper is to extend a work of Sivatski to characteristic 2. More precisely, for $F$ a field of characteristic $2$ and a central simple algebra $A$ of exponent 2 that splits over a triquadratic extension of $F$ of separability degree at least 4, we attach a cohomological invariant $\inv(A) \in H_2^3(F) / G$, where $H_2^3(F)$ is the third Kato-Milno cohomology group and $G$ is a subgroup of $H_2^3(F)$ divisible by the Brauer class of $A$. As an application, we will relate the decomposability of the algebra in degree 8 to the vanishing of $\inv(A)$. Moreover, we will use this invariant to prove some descent results for central simple algebras and quadratic forms over biquadratic extensions.