Symbol length in positive characteristic

TL;DR

This paper investigates the structure of certain central simple algebras in prime characteristic, providing bounds on their decomposition into cyclic algebras and improving known results for specific cases.

Contribution

It establishes new bounds on the number of cyclic algebras needed to represent Brauer classes of exponent p in prime characteristic, including improved results for p=2.

Findings

Any central simple algebra of exponent p split by a p-extension of degree p^n is Brauer equivalent to a tensor product of 2·p^{n-1}-1 cyclic algebras.

For p=2 and n≥3, such an algebra is Brauer equivalent to a tensor product of 5·2^{n-3}-1 quaternion algebras.

Provides new proofs for bounds on the minimal number of cyclic algebras representing Brauer classes in prime characteristic.

Abstract

We show that any central simple algebra of exponent $p$ in prime characteristic $p$ that is split by a $p$-extension of degree $p^n$ is Brauer equivalent to a tensor product of $2\cdot p^{n-1}-1$ cyclic algebras of degree $p$. If $p=2$ and $n\geq3$, we improve this result by showing that such an algebra is Brauer equivalent to a tensor product of $5\cdot2^{n-3}-1$ quaternion algebras. Furthermore, we provide new proofs for some bounds on the minimum number of cyclic algebras of degree $p$ that is needed to represent Brauer classes of central simple algebras of exponent $p$ in prime characteristic $p$, which have previously been obtained by different methods.