On the symbol length of -algebras

Abstract

The main result of this paper states that if k is a field of characteristic p > 0 and A/k is a central simple algebra of index d = p n and exponent p e , then A is split by a purely inseparable extension of k of the form k( p e √ a i , i = 1, . . . , d − 1). Combining this result with a theorem of Albert (for which we include a new proof), we get that any such algebra is Brauer equivalent to the tensor product of at most d − 1 cyclic algebras of degree p e . This gives a drastic improvement upon previously known upper bounds.

Symbol length in the Brauer group of a field

“…(7) Every p-algebra of index p n and exponent p m is similar to the product of p n − 1 cyclic algebras of degree p m . That is, len(p m , p n ) = p n − 1 (Florence [9]). (8) If F is the function field of an l-adic curve containing a primitive p-th root of one and p is a prime different than l, then every degree p algebra is cyclic.…”

“…It turns out that things are much simpler in this case, both for general fields as shown by Florence in [9] and for C m fields where things basically follow from an exercise in Serre, [25].…”

“…General fields of characteristic p. In [9] Florence proves the following theorem: Theorem 7.1. If F is a field with char(F ) = p and A is an F -csa of index p n and exponent p e , then A is similar to the product of at most p n − 1 cyclic algebras.…”

“…Here is a short proof along the lines of [9]. The idea is based on the following two well known theorems.…”

Motivic Decomposition of Projective Pseudo-Homogeneous Varieties

“…Proof. Note that B = A (p) by Theorem 3.9 in [26] (see also Proposition 3.2 in [14]). Therefore, The rest follows easily.…”