Sums of three integral squares in biquadratic fields (Q2017176)

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.