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A quadratic field which is Euclidean but not norm-Euclidean. (English) Zbl 0817.11047
The author uses earlier methods of E. S. Barnes and H. P. F. Swinnerton-Dyer [Acta Math. 87, 259-323 (1952; Zbl 0046.276)] to prove with the help of a computer that the ring $ℤ\left[\frac{1+\sqrt{69}}{2}\right]$ is Euclidean. This is the first example of a quadratic number field shown to be Euclidean but not norm-Euclidean.
Reviewer: M.Pohst (Berlin)

##### MSC:
 11R11 Quadratic extensions 11Y40 Algebraic number theory computations 11A05 Multiplicative structure of the integers
##### Keywords:
 [1] E.S. Barnes and H.P.F. Swinnerton-Dyer,The Inhomogeneous Minima of Binary Quadratic Forms, Acta Math.87 (1952), 259–323 · Zbl 0046.27601 · doi:10.1007/BF02392288 [2] D.A. Clark,The Euclidean Algorithm for Galois Extensions of the Rational Numbers, Ph.D. Thesis, McGill University, Montréal, 1992 [3] D. A. Clark and M.R. Murty, The Euclidean Algorithm in Galois Extensions of $ℚ$, (to appear) [4] L.E. Dickson,Algebren und ihre Zahlentheorie, Orell Füssli Verlag, Zürich und Leipzig, 1927 [5] P.G. Lejeune Dirichlet (ed. R. Dedekind),Vorlesungen über Zahlentheorie, Vieweg, Braunschweig, 1893 [6] O. Perron,Quadratische Zahlkörpern mit Euklidischem Algorithmus, Math. Ann.107 (1932), 489–495 · doi:10.1007/BF01448906 [7] P. Weinberger,On Euclidean Rings of Algebraic Integers, Proc. Symp. Pure Math.24 (1973), 321–332