Decomposable and Indecomposable Algebras of Degree 8 and Exponent 2

TL;DR

This paper investigates the decomposition of degree 8, exponent 2 central simple algebras into quaternion tensor products, introduces a cohomological invariant to determine descent, and constructs examples of indecomposable algebras over specific fields.

Contribution

It introduces a degree 3 cohomological invariant that detects descent of biquaternion algebras and constructs explicit indecomposable algebras of degree 8 and exponent 2.

Findings

The invariant determines whether a biquaternion algebra has a descent to the base field.

Examples of indecomposable algebras of degree 8 and exponent 2 are constructed over fields with specific cohomological properties.

Chow group computations are used to support the construction of indecomposable algebras.

Abstract

We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let $B$ be a biquaternion algebra over $F(\sqrt{a})$ with trivial corestriction. A degree 3 cohomological invariant is defined and we show that it determines whether $B$ has a descent to $F$. This invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2-cohomological dimension 3 and over a field $\mathbb M(t)$ where the $u$-invariant of $\mathbb M$ is 8 and $t$ is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by A. S. Merkurjev in Appendix.