  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
minumum 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.

Rings and Algebras