Real Quadratic Orders with Large Class Number

Acknowledgements I am undoubtedly indebted to my two supervisors, Alf van der Poorten and Hugh Williams. Most students would feel lucky to have one good supervisor whereas I had the privilege of having two excellent mathematicians to guide me. Their help, wisdom and infinite patience has been invaluable, and I am very grateful for it. The Mathematics Department at Macquarie University has provided a pleasant and enjoyable environment for the past few years. I would also like to thank both the University of Calgary and the University of Manitoba for their warm and generous hospitality during my visits to Canada. Those visits were made possible by an ARC IREX grant, and these visits were vital in the development of the thesis. Also, I owe a great debt to my Canadian hosts, Lynne Romuld in Winnipeg and Kris and Gardy Vasudevan in Calgary, who took me in and made me feel welcome. Last, but not least I would also like to thank all my family and friends, in particular my lovely wife Cath for all her patience and inspiration. Certificate This thesis is presented for the degree of Doctor of Philosophy at Macquarie University. It has not been submitted for a higher degree at any other university or institution.