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On the Symbol Length of Fields with finite Square Class Number

Detlev Hoffmann, Nico Lorenz

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 $φ\in \mathsf I^n(F)$ there exists some $0\leq k\leq \mathrm{sl}_n(F)$ and Pfister forms $π_1,\ldots, π_k$ such that $\varphi\equiv π_1+\ldots+π_k\mod \mathsf I^{n+1}(F)$. In particular, we rediscover a bound that can also be deduced from a result by Bruno Kahn that he stated without proof.

On the Symbol Length of Fields with finite Square Class Number

Abstract

Let be a field of characteristic not 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 . Moreover, we compute upper bounds for the -symbol length (), i.e., the smallest integer such that to each quadratic form there exists some and Pfister forms such that . In particular, we rediscover a bound that can also be deduced from a result by Bruno Kahn that he stated without proof.

Paper Structure

This paper contains 3 sections, 17 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. The Number of Pfister Forms
  3. Bounding the Symbol Length

Key Result

Proposition 2.1

We have

Theorems & Definitions (41)

  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Example 2.6
  • Definition 2.7
  • ...and 31 more