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Non-galois cubic fields which are euclidean but not norm-euclidean
Author(s):
David
A.
Clark.
Journal:
Math. Comp.
65
(1996),
1675-1679.
MSC (1991):
Primary 11A05;
Secondary 11R16
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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.
References:
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- E.S. Barnes and H.P.F. Swinnerton-Dyer, The inhomogeneous minima of binary quadratic forms, Acta Math. 87 (1952), 259--323. MR 14:730a
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- D.A. Clark, The Euclidean algorithm for Galois extensions of the rational numbers, McGill University, Montréal, 1992.
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- ------, A quadratic field which is Euclidean but not norm-Euclidean, Manuscripta Math. 83 (1994), 327--330. MR 95f:11086
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- D.A. Clark and M.R. Murty, The Euclidean algorithm in Galois extensions of
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- F. Lemmermeyer, The Euclidean algorithm in algebraic number fields, Exposition. Math. 13 (1995), 385--416. CMP 96:04
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- H.W. Lenstra, Lectures on Euclidean rings, Bielefeld, 1974.
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- 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.
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- P. Weinberger, On Euclidean rings of algebraic integers, Proc. Symp. Pure Math. 24 (1973), 321--332. 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:
10.1090/S0025-5718-96-00764-8
PII:
S 0025-5718(96)00764-8
Received by editor(s):
February 18, 1994
Received by editor(s) in revised form:
April 15, 1995 and August 11, 1994 and February 22, 1995
Copyright of article:
Copyright
1996,
American Mathematical Society
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