On the Symbol Length of Fields with finite Square Class Number

Abstract

Let $F$ be a field of characteristic not $2$ with finitely many square classes. Using combinatorial arguments applied to objects related to vector spaces over finite fields, we deduce an upper bound for the number of Pfister forms over $F$. Moreover, we compute upper bounds for the $n$-symbol length $F$ ($n\in \mathbb{N}$), i.e., the smallest integer ${\mathrm{sl}}_n(F)\ge 0$ such that to each quadratic form $\varphi \in {\mathsf{I}}^n(F)$ there exists some $0\le k\le {\mathrm{sl}}_n(F)$ and Pfister forms ${\pi }_1,\dots ,{\pi }_k$ such that $\phi \equiv {\pi }_1+\dots +{\pi }_k\mathrm{mod}{\mathsf{I}}^{n+1}(F)$. In particular, we rediscover a bound that can also be deduced from a result by Bruno Kahn that he stated without proof.