manuscripta mathematica

, Volume 83, Issue 1, pp 327-330

First online:

A quadratic field which is Euclidean but not norm-Euclidean

  • David A. ClarkAffiliated withInstitut für Experimentelle Mathematik, Universität GHS EssenDepartment of Mathematics, Brigham Young University

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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