Invenüones math 38 237-254(1977) ^ mathematicae © by Springer Verlag 1977 Euclidean Number Fields of Large Degree H W Lenstra, Jr Mathematisch Instituut Umversiteit van Amsterdam Roetersstraat 15 Amsterdam The Netherlands Introduction Lei K be a number field, and let R be the img of algebraic mtegers m K We say thal K is Euclidean, or thal R is Euclidean with respect to the norm, if for every a,beR, b + 0, 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 contams many elements all of whose differences are units, see Section l for details Some remarks about the existence of such elements are made m Section 2 In Section 3 we illustrate the method by givmg 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 foi commumcatmg 10 me the results of [35], and to P van Emde Boas, AK Lenstra and R H Mak for their help m computmg disciimmants §1. A Sutficient Condition for Euclid's Algorithm In this section K denoies an algebraic number field of finite degree n and discnmmantzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ΔzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA over tbe field of rational numbers Q By r and s we mean the number of real and complex archimedean pnmes of K, respectively The ring of algebraic mtegers m K is denoted by R We legard K äs bemg embedded m the R-algebra K K = K (x) Q R, which, äs an R-algebra, is isomorphic to R r χ C s As an R vector space we identify C  with  R 2  by sendmg a + bi to (a + b, a — b),  for a, beR This leads to an Identification of K R =R r xC s  with  the n dimensional Euclidean  space R" It  is well known  that  this Identification  makes R mto a lattice of determmant \ A\ 3  m R"