Euclidean real quadratic number fields (Q1057932)

Let \(K\) be a real quadratic field and let \({\mathcal O}_ K\) denote the ring of integers of \(K\). The authors call \(K\) ``Generalized Euclidean'' (= G.E.) iff given \(\alpha\),\(\beta\in {\mathcal O}_ K\), \(\beta\neq 0\) and the ideal \((\alpha,\beta)\) principal, there exist \(\gamma,\delta\in {\mathcal O}_ K\) such that \(\alpha =\beta \delta +\gamma\), \(| N(\gamma)| <| N(\beta)|\). The following results are proved: (1) If \(d>1\) is square-free and \(d\not\equiv 1 \bmod 4\), then \(K=\mathbb Q(\sqrt{d})\) is G.E. only if \(K\) is Euclidean (with respect to the norm) except when \(d=10\). (2) If \(K=\mathbb Q(\sqrt{p})\), where \(p\) ranges over all primes \(\equiv 1\bmod 4\), then \(K\) is G.E. for only finitely many \(p\).