On the Symbol Length of Fields with finite Square Class Number

TL;DR

This paper establishes upper bounds on the symbol length of quadratic forms over fields with finitely many square classes, using combinatorial methods related to vector spaces over finite fields.

Contribution

It provides new bounds for the n-symbol length of quadratic forms over such fields, including a rediscovery of a known bound via a different approach.

Findings

Derived an upper bound for the number of Pfister forms over the field.

Computed upper bounds for the n-symbol length for fields with finitely many square classes.

Reproduced a known bound originally stated by Bruno Kahn.

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 $\mathrm{sl}_n(F)\geq 0$ such that to each quadratic form $\phi\in \mathsf I^n(F)$ there exists some $0\leq k\leq \mathrm{sl}_n(F)$ and Pfister forms $\pi_1,\ldots, \pi_k$ such that $\varphi\equiv \pi_1+\ldots+\pi_k\mod \mathsf I^{n+1}(F)$. In particular, we rediscover a bound that can also be deduced from a result by Bruno Kahn that he stated without proof.