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 \(\mathbb{Z}[ {{1+ \sqrt {69}} \over 2}]\) is Euclidean. This is the first example of a quadratic number field shown to be Euclidean but not norm-Euclidean.
Reviewer: M.Pohst (Berlin)


11R11 Quadratic extensions
11Y40 Algebraic number theory computations
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors


Zbl 0046.276
Full Text: DOI EuDML


[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
[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 \(\mathbb{Q}\), (to appear) · Zbl 0814.11049
[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 · Zbl 0376.10001
[6] O. Perron,Quadratische Zahlkörpern mit Euklidischem Algorithmus, Math. Ann.107 (1932), 489–495 · Zbl 0005.38703
[7] P. Weinberger,On Euclidean Rings of Algebraic Integers, Proc. Symp. Pure Math.24 (1973), 321–332 · Zbl 0287.12012
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