On the norm-Euclideanity of \(\mathbb Q\left(\sqrt{2+\sqrt{2+\sqrt 2}}\right)\) and \(\mathbb Q\left(\sqrt{2+\sqrt 2}\right)\) (Q5939699)

The norm-Euclideanity of a field \(K\) means that given a rational \(\gamma\) in \(K\) there exists an integer \(\alpha\) in \(K\) such that \(|N(\alpha-\gamma)|\leq M(K)< 1\). This is notably most difficult for \(K\) real [see \textit{F. Lemmermeyer}, Expo. Math. 13, 385--416 (1995; Zbl 0843.11046)]. NEWLINENEWLINENEWLINEThe computational method is to restrict \(\gamma\) to a fundamental region for integral translation, and to show for a finite set of \(\alpha\) the norm inequality holds. Thus it is classical that for \(K= \mathbb Q(\sqrt{2})\), \(M=1/2\), with (one) \(\alpha=0\). NEWLINENEWLINENEWLINEThe reviewer and \textit{J. Deutsch} showed [Math. Comput. 46, 295--299 (1986; Zbl 0585.12002)] that for \(K= \mathbb Q\left(\sqrt{2+ \sqrt{2}}\right)\), there is a larger set of \(\alpha\), but \(M<1\) and conjecturally \(1/2\) again (as shown by the author using four values of \(\alpha\)). NEWLINENEWLINENEWLINEHe also shows that the same holds for \(\mathbb Q \left(\sqrt{2+ \sqrt{2+ \sqrt{2}}}\right)\), using 20 values of \(\alpha\).