On the Number of Square Classes of a Field of Finite Level

The level question is, whether there exists a field F with finite square class number q(F): = |F × /F ×2 | and finite level s(F) greater than four. While an answer to this question is still not known, one may ask for lower bounds for q(F) when the level is given. For a nonreal field F of level s(F) = 2 n, we consider the filtration of the groups DF (2 i), 0 ≤ i ≤ n, consisting of all the nonzero sums of 2 i squares in F. Developing further ideas of A. Pfister, P. L. Chang and D. Z. Djoković and by the use of combinatorics, we obtain lower bounds for the invariants q i: = |DF (2 i)/DF (2 i−1)|, for 1 ≤ i ≤ n, in terms of s(F). As a consequence, a field with finite level ≥ 8 will have at least 512 square classes. Further we give lower bounds on the cardinalities of the Witt ring and of the 2-torsion part of the Brauer group of such a field