## 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

### Keywords:

Euclidean fields; norm-Euclideanity

### Citations:

Zbl 0843.11046; Zbl 0985.12002; Zbl 0585.12002
Full Text:

### 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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