Sums of three integral squares in biquadratic fields
type: Item
instance of: scholarly article
title: Sums of three integral squares in biquadratic fields
zbMATH Open document ID: 1294.11044
author: Bin Zhang
author: Chungang Ji
published in: Journal of Number Theory
publication date: 2014-06-25T00:00:00Z
review text: This article deals with sums of squares in number fields \(K\). Let \({\mathcal O}_K\) denote its ring of integers. The stufe \(s({\mathcal O}_K)\) is defined as the smallest number \(n\) such that \(-1\) is a sum of \(n\) squares in \({\mathcal O}_K\), and \(S_K\) is the set of all elements in \({\mathcal O}_K\) that can be written as sums of squares. Then the following results are proved: Let \(K = \mathbb Q(\sqrt{-m},\sqrt{-n}\,)\) for positive squarefree integers \(m \equiv n \equiv 3 \bmod 4\); then \(S_K = {\mathcal O}_K\). If, in addition, \({\mathcal O}_K\) has stufe \(2\), then every element in \(S_K\) can be written as a sum of at most three squares. The proofs are elementary and use the solvability of the Pell equation.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 6308436
zbMATH Keywords: biquadratic number field
zbMATH Keywords: sum of squares
zbMATH Keywords: ring of integers
zbMATH Keywords: stufe
zbMATH Keywords: Pell equation
MaRDI profile type: Publication
full work available at URL: j.jnt.2013.11.013
OpenAlex ID: W1967817170
cites work: Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms
cites work: Sums of Three Integer Squares in Complex Quadratic Fields
cites work: Sums of three squares over imaginary quadratic fields
cites work: Q5616966
cites work: Q5607309
cites work: Sums of \(m\)th powers of algebraic integers
DOI: 10.1016/J.JNT.2013.11.013
Recommended article: Sums of Three Squares and Levels of Quadratic Number Fields
Recommended article: Q4696507
Recommended article: Q3362757
Recommended article: Sums of three squares over imaginary quadratic fields
Recommended article: Sums of three integral squares in cyclotomic fields
Recommended article: Sum of three squares and class numbers of imaginary quadratic fields
Recommended article: On the sum of two integral squares in certain quadratic fields
Recommended article: Sums of distinct integral squares in real quadratic fields
Recommended article: Sums of nonvanishing integral squares in real quadratic fields
Recommended article: Q3697094
description: scientific article; zbMATH DE number 6308436