On the common slot property for symbol algebras

Abstract

Let k be a field, let n≥2 be a nonsquarefree integer not divisible by the characteristic of k. Assume that all roots of unity of degree n are contained in k. In the first part of the paper we consider pairs of symbol algebras over k with common slots D1≃(e,x)n≃(r,u)n, D2≃(e,y)n≃(r,v)n, expD1=expD2=n, and show that in general (e,x,y)n≠(r,u,v)n. As a consequence we prove that in general it is impossible to connect the pair {(e,x)n;(e,y)n} and the pair {(r,u)n;(r,v)n} by a chain of pairs of symbol algebras with a common slot and isomorphic to (D1;D2) in such a way that any two neighboring pairs in the chain are obtained from one another by a “natural” transformation.In the second part of the paper we prove that in contrast to the case n=2 for any n divisible by 4 there exist symbol algebras D1, D2 with deg⁡D1=deg⁡D2=n and expD1=expD2=n without common slot such that iD1+jD2 is a symbol algebra of degree n for any i,j∈Z.