Algorithms in algebraic number theory

"Lenstra formulates the basic problems of algorithmic algebraic number theory in rigorous terms, discussing advances and unsolved questions. He shows that the study of algorithms not only increases understanding of algebraic number fields, but also stimulates curiosity about them."@en
"Algorithms in algebraic number theory are as old as the field itself. Traditionally, the users of such algorithms were number theorists needing to do computations in algebraic number fields. However, recent applications, such as factoring large integers, have changed this situation. Lenstra presents a clear, well-paced, and fascinating lecture on some of the fundamental questions arising in this area. He formulates the basic problems of algorithmic algebraic number theory in rigorous terms, discussing advances and unsolved questions. The main topic of the lecture is the investigation of the multiplicative structure of rings of algebraic integers, the principal tool being a group that simultaneously describes the class group and the group of units of such a ring. Lenstra shows that the study of algorithms not only increases understanding of algebraic number fields, but also stimulates curiosity about them. For this reason, the lecture would be an excellent addition to a course touching on these topics. It is accessible to advanced undergraduates and graduate students with backgrounds in algebra and number theory."