Quaternions and universal quadratic forms over number fields

Matěj Doležálek

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arXiv · 2012.07097

Quaternions and universal quadratic forms over number fields

Matěj Doležálek

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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.

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Matěj Doležálek. 2020-12-13. Quaternions and universal quadratic forms over number fields. https://arxiv.org/abs/2012.07097

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