P-Algebras over an algebraic function field over a perfect field

Abstract

Let K be a field of characteristic p > 0. Suppose that K is an algebraic function field of r variables over a perfect field. We shall consider the structure of p-algebras over K. When r = 1, Albert proved that every p-algebra is, in fact, a cyclic algebra and the exponent is equal to the index [a]. In this note we shall generalize Albert’s result to the following: Let K be an algebraic function field of r variables over a perfect field, A any p-algebra over K. Then there are cyclic division algebras D, , D2,..., D, so that A is similar to D, OK D, ok ... ok D,, the exponent of each Di is equal to its index, and the exponent of each Di is no greater than that of A. In fact a more general theorem can be established. I thank Professor Shuen Yuan and the referee for their very helpful suggestions about revising my original manuscripts. Throughout this note a cyclic algebra is denoted by (a, L/K, a) where 0 is a generator of Gal(L/K). The Brauer classes of a central simple K-algebra A and a cyclic algebra (a, L/K, a) will be denoted by [A] and [a, L/K, 01, respectively. o(A) is the exponent of A, i(A) its index and deg A = Jm its degree. The following easy lemma is from field theory. We omit its proof.