Abstract.
The level of a ring R with 1 ≠ 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 2n or 2n + 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.
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Supported in part by the European research network HPRN-CT-2002-00287 “Algebraic K-Theory, Linear Algebraic Groups and Related Structures”.
Received: 23 March 2007, Revised: 30 October 2007
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Hoffmann, D.W. Levels of quaternion algebras. Arch. Math. 90, 401–411 (2008). https://doi.org/10.1007/s00013-008-2380-y
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DOI: https://doi.org/10.1007/s00013-008-2380-y