LEVELS AND SUBLEVELS OF QUATERNION ALGEBRAS

Abstract

The level s (resp. sublevel s) of a ring R with 1 6= 0 is the smallest positive integer such that −1 (resp. 0) can be written as a sum of s (resp. s+1) nonzero squares in R, provided −1 (resp. 0) is a sum of nonzero squares at all. D.W. Lewis showed that any value of type 2n or 2n +1 can be realized as level of a quaternion division algebra, and in all these examples, the sublevel was 2n, which prompted the question whether or not the level and sublevel of a quaternion division algebra will always differ at most by one. In this note, we give a positive answer to that question.

Locations

• Mathematical Proceedings of the Royal Irish Academy - View
• CiteSeer X (The Pennsylvania State University) - View - PDF