Symbol length in positive characteristic

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 center dot p(n-1) -1 cyclic algebras of degree p. If p = 2 and n >= 3, we improve this result by showing that such an algebra is Brauer equivalent to a tensor product of 5 center dot 2n-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. (c) 2024 Elsevier B.V. All rights reserved.