manuscripta mathematica

, Volume 83, Issue 1, pp 327–330 | Cite as

A quadratic field which is Euclidean but not norm-Euclidean

  • David A. Clark


The classification of rings of algebraic integers which are Euclidean (not necessarily for the norm function) is a major unsolved problem. Assuming the Generalized Riemann Hypothesis, Weinberger [7] showed in 1973 that for algebraic number fields containing infinitely many units the ring of integersR is a Euclidean domain if and only if it is a principal ideal domain. Since there are principal ideal domains which are not norm-Euclidean, there should exist examples of rings of algebraic integers which are Euclidean but not norm-Euclidean. In this paper, we give the first example for quadratic fields, the ring of integers of\(\mathbb{Q}\left( {\sqrt {69} } \right)\).

1991 Mathematics Subject Classification

Primary 11A05 Secondary 11R16 


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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • David A. Clark
    • 1
    • 2
  1. 1.Institut für Experimentelle MathematikUniversität GHS EssenEssenGermany
  2. 2.Department of MathematicsBrigham Young UniversityProvoUSA

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