Symbol length of p-algebras of prime exponent

Abstract

We prove that if the maximal dimension of an anisotropic homogeneous polynomial form of prime degree [Formula: see text] over a field [Formula: see text] with [Formula: see text] is a finite integer [Formula: see text] greater than 1 then the symbol length of [Formula: see text]-algebras of exponent [Formula: see text] over [Formula: see text] is bounded from above by [Formula: see text], and show that every two tensor products of symbol algebras of lengths [Formula: see text] and [Formula: see text] with [Formula: see text] can be modified so that they share a common slot. For [Formula: see text], we obtain an upper bound of [Formula: see text] for the symbol length, which is sharp when [Formula: see text].