Quaternions and universal quadratic forms over number fields

2020/12/13 by Matěj Doležálek, Doležálek, Matěj
Mathematics · #(11R16) #11E12 #11H06 #11R52 #11R80 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2012.07097

openalex publication_date 2020/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study quadratic forms over totally real number fields by using an associated ring of quaternions. We examine some properties of residue class rings of these quaternions and use geometry of numbers to prove that certain ideals of the ring of quaternions contain elements of a small norm. We prove that x²+y²+z²+w²+xy+xz+xw is universal over ℚ(ζ₇+ζ₇^-1) and that x²+xy+y²+z²+zw+w² represents all totally positive multiples of certain special elements.

Citations

• A Quaternionic Proof of the Universality of Some Quadratic Forms

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