Nongalois cubic fields which are euclidean but not normeuclidean
Author:
David A. Clark
Journal:
Math. Comp. 65 (1996), 16751679
MSC (1991):
Primary 11A05; Secondary 11R16
MathSciNet review:
1355007
Fulltext PDF Free Access
Abstract 
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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 normEuclidean.
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 E.S. Barnes and H.P.F. SwinnertonDyer, The inhomogeneous minima of binary quadratic forms, Acta Math. 87 (1952), 259323. MR 14:730a
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 [5]
 F. Lemmermeyer, The Euclidean algorithm in algebraic number fields, Exposition. Math. 13 (1995), 385416. 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. 223224.
 [8]
 E.M. Taylor, Euclid's algorithm in cubic fields with complex conjugates, J. London Math. Soc. 14 (1976), 4954. MR 54:7420
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 P. Weinberger, On Euclidean rings of algebraic integers, Proc. Symp. Pure Math. 24 (1973), 321332. MR 49:2671
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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/S0025571896007648
PII:
S 00255718(96)007648
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
