Malcolm Harper - A family of Euclidean rings containing

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Malcolm Harper - A family of Euclidean rings containing $Z [\sqrt{14}]$

MALCOLM HARPER, Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 2K6, Canada

A family of Euclidean rings containing $Z [\sqrt{14}]$

Let K be an algebraic number field with ring of integers ${\cal O}_{K}$ and suppose that ${\cal O}_{K}$ has an infinite unit group. Assumming a suitable generalized Riemann hypothesis, ${\cal O}_{K}$ is a Euclidean ring (in the sense of Samuel, 1971) if and only if K has class number 1 (Weinberger, 1973). $Z [\sqrt{14}]$ has an infinite unit group and is the ring of integers in $K=Q (\sqrt{14})$ which has class number 1. Cardon (1997) showed that the fundamental obstruction to the norm acting as a Euclidean algorithm in $Z [\sqrt{14}]$ lies at one of the residue classes modulo 2 and thus $Z[\sqrt{14}] [1/2]$ is Euclidean. Using the sieve techniques of Gupta, Murty and Murty (1987), Clark (1992) and Clark and Murty (1995) we show that inverting any non-unit in $Z [\sqrt{14}]$ yields a Euclidean ring. That is, $Z [\sqrt{14}] [1/a]$ is Euclidean for any a in $Z [\sqrt{14}]$ not a unit.

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