Euclidean number fields of large degree

Let K be a number field, and let R be the ring of algebraic integers in K. We say that K is Euclidean, or that R is Euclidean with respect to the norm. if for every a, beR, bq=O, there exist c, deR such that a=cb+d and N(d)<N(b). Here N denotes the absolute value of the field norm K--+Q. This paper deals with a new technique of proving fields to be Euclidean. The method, which is related to an old idea of Hurwitz [14], is based on the observation that for K to be Euclidean it suffices that R contains many elements all of whose differences are units; see Section 1 for details. Some remarks about the existence of such elements are made in Section 2. In Section 3 we illustrate the method by giving 132 new examples of Euclidean fields of degrees four, five, six, seven and eight. A survey of the known Euclidean fields is given in Section 4. Acknowledgements are due to B. Matzat for making available [1] and [23]; to E.M. Taylor for communicating to me the results of [35] ; and to R van Emde Boas, A.K. Lenstra and R.H. Mak for their help in computing discriminants.