A refinement of the Dress-Scharlau theorem (Q495289)

Let \(K_d=\mathbb{\mathbb Q}(\sqrt{d})\) be a real quadratic field. For \(\alpha=a+b\sqrt{d} \in K_d\), let \(\overline{\alpha}=a-b\sqrt{d}\) be the conjugate of \(\alpha\). Let \(\mathcal{O}_{d}^{\pm}\) be the set of \(\alpha \in \mathcal{O}_d\) satisfying \(\alpha>0\) and \(\overline{\alpha}<0\). The authors show that if \(\alpha \in \mathcal{O}_{d}^{\pm}\) and the norm \(|\alpha \overline{\alpha}|\) is greater than \(d\) if \(d \equiv 2,3 \pmod 4\) (resp. greater than \((d-1)/4\) if \(d \equiv 1 \pmod 4\)) then \(\alpha\) is decomposable in \(\mathcal{O}_{d}^{\pm}\), i.e., there exist \(\beta\) and \(\gamma \in \mathcal{O}_{d}^{\pm}\) such that \(\alpha=\beta+\gamma\) (Theorem 4). Other results of this type are also obtained.