Detlev W. Hoffmann: Levels of Quaternion Algebras

Submission: 2007, Jan 31

The level of a ring $R$ with $1\neq 0$ is the smallest positive integer $s$ such that $-1$ can be written as a sum of $s$ squares in $R$, provided $-1$ is a sum of squares at all. D.W. Lewis showed that any value of type $2^n$ or $2^n+1$ can be realized as level of a quaternion algebra, and he asked whether there exist quaternion algebras whose levels are not of that form. Using function fields of quadratic forms, we construct such examples.

2000 Mathematics Subject Classification: Primary 11E04; Secondary 11E25, 12D15, 16K20

Keywords and Phrases: Quaternion algebra, level, sublevel, quadratic form, function field of a quadratic form

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