Non-galois cubic fields which are euclidean but not norm-euclidean

Author:
David A. Clark

Journal:
Math. Comp. **65** (1996), 1675-1679

MSC (1991):
Primary 11A05; Secondary 11R16

MathSciNet review:
1355007

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Abstract: Weinberger in 1973 has shown that under the Generalized Riemann Hypothesis for Dedekind zeta functions, an algebraic number field with infinite unit group is Euclidean if and only if it is a principal ideal domain. Using a method recently introduced by us, we give two examples of cubic fields which are Euclidean but not norm--Euclidean.

**[1]**E. S. Barnes and H. P. F. Swinnerton-Dyer,*The inhomogeneous minima of binary quadratic forms. I*, Acta Math.**87**(1952), 259–323. MR**0053162****[2]**D.A. Clark,*The Euclidean algorithm for Galois extensions of the rational numbers*, McGill University, Montréal, 1992.**[3]**David A. Clark,*A quadratic field which is Euclidean but not norm-Euclidean*, Manuscripta Math.**83**(1994), no. 3-4, 327–330. MR**1277533**, 10.1007/BF02567617**[4]**D.A. Clark and M.R. Murty,*The Euclidean algorithm in Galois extensions of*, J. Reine Angew. Math.**459**(1995), 151--162. CMP**95:09****[5]**F. Lemmermeyer,*The Euclidean algorithm in algebraic number fields*, Exposition. Math.**13**(1995), 385--416. CMP**96:04****[6]**H.W. Lenstra,*Lectures on Euclidean rings*, Bielefeld, 1974.**[7]**J.R. Smith,*The inhomogeneous minima of some totally real cubic fields*, Computers in Number Theory (A.O.L. Atkin and B.J. Birch, eds.), Academic Press, New York, 1971, pp. 223--224.**[8]**Elizabeth M. Taylor,*Euclid’s algorithm in cubic fields with complex conjugates*, J. London Math. Soc. (2)**14**(1976), no. 1, 49–54. MR**0419399****[9]**Peter J. Weinberger,*On Euclidean rings of algebraic integers*, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R. I., 1973, pp. 321–332. MR**0337902**

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Additional Information

**David A. Clark**

Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602

Email:
clark@math.byu.edu

DOI:
http://dx.doi.org/10.1090/S0025-5718-96-00764-8

Received by editor(s):
February 18, 1994

Received by editor(s) in revised form:
April 15, 1995, August 11, 1994, and February 22, 1995

Article copyright:
© Copyright 1996
American Mathematical Society