Quaternions and universal quadratic forms over number fields

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 $x^2+y^2+z^2+w^2+xy+xz+xw$ is universal over $\mathbb{Q}({\zeta }_7+{\zeta }_7^{-1})$ and that $x^2+xy+y^2+z^2+zw+w^2$ represents all totally positive multiples of certain special elements.

References in corpus (5)

• Number fields without n-ary universal quadratic forms
• Universal quadratic forms and elements of small norm in real quadratic fields
• Lifting problem for universal quadratic forms
• Universal quadratic forms over multiquadratic fields
• A Quaternionic Proof of the Universality of Some Quadratic Forms