Quaternary universal forms over \(\mathbb Q(\sqrt{13})\)
type: Item
instance of: scholarly article
title: Quaternary universal forms over \(\mathbb Q(\sqrt{13})\)
zbMATH Open document ID: 1193.11033
author: Hideyo Sasaki
published in: The Ramanujan Journal
publication date: 2009-11-23T00:00:00Z
review text: Let \(F=\mathbb{Q}(\sqrt{m})\) be a real quadratic field with the ring of integers \(\mathcal{O}\). A totally positive definite integral \(n\)-ary quadratic form \(f=f(x_1,\dots,x_n)\) with coefficients in \(\mathcal{O}\) is called \textit{universal} if it represents all totally positive integers in \(\mathcal{O}\). The author proves that there are only two quaternary universal forms (up to equivalence) over \(\mathbb{Q}(\sqrt{13})\): \[ x_1^2+x_2^2+\frac{5+\sqrt{13}}{2}x_3^2+\frac{5-\sqrt{13}}{2}x_4^2+2x_3x_4 \] and \[ x_1^2+2x_2^2+\frac{5+\sqrt{13}}{2}x_3^2+\frac{5-\sqrt{13}}{2}x_4^2+2x_2x_3+2x_2x_4. \]
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11E20
zbMATH DE Number: 5636938
zbMATH Keywords: quaternary quadratic forms
zbMATH Keywords: quadratic fields
MaRDI profile type: Publication
full work available at URL: s11139-007-9056-2
OpenAlex ID: W2020484120
cites work: Q3125857
cites work: On the class number of a unit lattice over a ring of real quadratic integers
DOI: 10.1007/S11139-007-9056-2
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description: scientific article; zbMATH DE number 5636938