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

The \emph{level question} is, whether there exists a field $F$ with finite \emph{square class number} $q(F):=|F^\times/{F^\times}^2|$ 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 $D_F(2^i)$, $0\leq i\leq n$, consisting of all the nonzero sums of $2^i$ squares in $F$. Developing further ideas of A. Pfister, P. L. Chang and D. Z. Djokovi'{c} and by the use of combinatorics, we obtain lower bounds for the invariants $\overline{q}_i:=|D_F(2^i)/D_F(2^{i-1})|$, for $1\leq i \leq n$, in terms of $s(F)$. As a consequence, a field with finite level $\geq 8$ 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.

2000 Mathematics Subject Classification:

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