Detlev W. Hoffmann: Levels and sublevels of quaternion algebras

Submission: 2009, Aug 6

The level (resp. sublevel) of a division ring R is the smallest positive integer m such that -1 (resp. 0) can be written as a sum of m (resp. m+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 2^n or 1+2^n can be realized as level of a quaternion division algebra, and in all these examples, the sublevel was 2^n, 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.

2000 Mathematics Subject Classification: 11E25, 16K20

Keywords and Phrases: level, sublevel, quaternion algebra, division algebra

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