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On the norm-Euclideanity of \(\mathbb Q\left(\sqrt{2+\sqrt{2+\sqrt 2}}\right)\) and \(\mathbb Q\left(\sqrt{2+\sqrt 2}\right)\). (De l’euclidianité de \(\mathbb Q\left(\sqrt{2+\sqrt{2+\sqrt 2}}\right)\) et \(\mathbb Q\left(\sqrt{2+\sqrt 2}\right)\) pour la norme.) (French) Zbl 1014.11064

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 F. Lemmermeyer, Expo. Math. 13, 385–416 (1995; Zbl 0843.11046)].
The 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) \(\alpha=0\).
The reviewer and 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 \(\alpha\)).
He also shows that the same holds for \(\mathbb Q \left(\sqrt{2+ \sqrt{2+ \sqrt{2}}}\right)\), using 20 values of \(\alpha\).

MSC:

11R18 Cyclotomic extensions
11Y40 Algebraic number theory computations
13F07 Euclidean rings and generalizations
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References:

[1] Samuel, P., Théorie algébrique des nombres. Hermann, Paris, 1971. · Zbl 0239.12001
[2] Borevitch, Z.I., Safarevitch, I.R., Théorie des nombres. Gauthier-Villars, Paris, 1967. · Zbl 0145.04901
[3] Washington, L.C., Introduction to cyclotomic fields. Graduate Texts in Mathematics, Springer-Verlag, New-York, 1982. · Zbl 0484.12001
[4] Lemmermeyer, F., The euclidean algorithm in algebraic number fields. Expositiones Mathematicae (1995), 385-416. · Zbl 0843.11046
[5] Cohn, H., A numerical study of Weber’s real class number calculation. Numerische Mathematik2 (1960), 347-362. · Zbl 0117.27501
[6] Cohn, H., Deutsch, J., Use of a computer scan to prove Q (√2+√2) and Q(√3+√2) are euclidean. Mathematics of Computation46 (1986), 295-299. · Zbl 0585.12002
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