On Euclidean Ideal Classes.

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

[发布机构] University of Michigan

[效力级别] Large Sieve

[关键词] Euclidean Ideal Class;Large Sieve;Gupta-Murty Bound;Euclidean;Class Group;Cyclic;Mathematics;Science;Mathematics