On Euclidean Ideal Classes.

Euclidean Ideal Class;Large Sieve;Gupta-Murty Bound;Euclidean;Class Group;Cyclic;Mathematics;Science;Mathematics

Graves, Hester K.Skinner, Christopher M. ;

University of Michigan

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

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/63828/gravesh_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 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.