Euclid's algorithm in cubic fields with complex conjugates

Abstract

This thesis investigates the existence of a Euclidean Algorithm in cubic fields with complex conjugates. This investigation is made using the following methods.The first method is a modification of a result of Cassels which states that no field of discriminant less than [equation] possesses a Euclidean Algorithm.By using the modification it is possible to show that some fields of discriminant greater then the above bound, but close to it, also do not possess a Euclidean Algorithm. A second method is to choose an algebraic integer which is a divisor, where is the fundamental unit of the field in question and n is a rational integer. We then determine whether there are any residue classes modulo p which do not contain an integer of norm of absolute value less than the absolute value of the norm.The next method is an adaptation of a method of Barnes and Swinnerton-Dyer for the real quadratic fields, modified here for the fields in question. This method aims to isolate the points with minimum at least 1. An indirect method, which is used as the final step of the last method described, is to determine the minimum of numbers of the form [equation], where [alpha] is an integer of the field in questionand n is a positive rational integer. In addition to existing results, 37 fields have been shownto possess a Euclidean Algorithm and it has been established that there is no Euclidean Algorithm in 289 fields. For some fields the inhomogeneous minimum has been determined. The numerical results obtained are given in the last chapter of this work. The listings of the computer programs used for the above methods are in the appendix to this thesis.