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. Otherwise, the level is
defined to be infinite.
A famous result by Pfister (1965) states that the level of a field, if finite,
is always a 2-power, and that each 2-power can be realized as level of a
suitable field. This answered a question asked by Van der Waerden in the 1930s.
In contrast, every positive
integer can be realized as level of an integral domain as was shown by Dai, Lam
and Peng (1980).
D.W. Lewis (1987) 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. We give a positive answer to that question by constructing
examples using methods from the theory of function fields of quadratic forms.