|MALCOLM HARPER, Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 2K6, Canada|
|A family of Euclidean rings containing|
Let K be an algebraic number field with ring of integers and suppose that has an infinite unit group. Assumming a suitable generalized Riemann hypothesis, is a Euclidean ring (in the sense of Samuel, 1971) if and only if K has class number 1 (Weinberger, 1973). has an infinite unit group and is the ring of integers in which has class number 1. Cardon (1997) showed that the fundamental obstruction to the norm acting as a Euclidean algorithm in lies at one of the residue classes modulo 2 and thus 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 yields a Euclidean ring. That is, is Euclidean for any a in not a unit.