On Euclidean Ideal Classes.

Abstract

In 1979, H.K.Lenstra generalized the idea of Euclidean algorithms to Euclidean ideal classes. If a domain has a Euclidean algorithm, then it is a principal ideal domain and has a trivial class group; if a Dedekind domain has a Euclidean ideal class, then it has a cyclic class group gen- erated by the Euclidean ideal class. Lenstra showed that if one assumes the generalized Riemann hypothesis and a number field has a ring of in- tegers with infinitely many units, then said ring has cyclic class group if and only if it has a Euclidean ideal class. Malcolm Harper’s dissertation built up general machinery that allows one to show a given ring of integers (with infinitely many units) of a number field with trivial class group is a Euclidean ring. In order to build the machinery, Harper used the Large Sieve and the Gupta-Murty bound. This dissertation generalizes Harper’s work to the Euclidean ideal class setting. In it, there is general machinery that allows one to show that a number field with cyclic class group and a ring of integers with infinitely many units has a Euclidean ideal class. In order to build this machinery, the Large Sieve and the Gupta-Murty bound needed to be generalized to the ideal class situation. The first required class field theory; the second required several asymptotic results on the sizes of sets of k-tuples.