On the Symbol Length of Fields with finite Square Class Number

2021/01/29 by Detlev W. Hoffmann, Hoffmann, Detlev, Nico Lorenz +1
Computer Science · Mathematics · #11E04 #11E81 #12D15 #12G05 #19D45 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2101.12593

openalex publication_date 2021/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈\mathbb N), i.e., the smallest integer sl_n(F)≥ 0 such that to each quadratic form ϕ∈ \mathsf Iⁿ(F) there exists some 0≤ k≤ sl_n(F) and Pfister forms π₁,…, π_k such that φ≡ π₁+…+π_k\mod \mathsf Iⁿ⁺¹(F). In particular, we rediscover a bound that can also be deduced from a result by Bruno Kahn that he stated without proof.

Related

• Pfister Numbers over Rigid Fields
• Chain Lemma, Quadratic Forms and Symbol Length
• Symbol Length of p-Algebras of Prime Exponent
• On the Pfister Number of Quadratic Forms
• Biquaternion Algebras, Chain Lemma and Symbol Length