Since arts are more easily learnt by examples than precepts, I have
thought fit to adjoin the solutions of the following problems.
Isaac Newton, in Universal Arithmetick
Learning is a mysterious process. No one can say what the precise rules
of learning are. However, it is an agreed upon fact that the study of good
examples plays a fundamental role in learning. With respect to mathematics, it is well-known that problem-solving helps one acquire routine skills in
how and when to apply a theorem. It also allows one to discover nuances of
the theory and leads one to ask further questions that suggest new avenues
of research. This principle resonates with the famous aphorism of Lichtenberg, “What you have been obliged to discover by yourself leaves a path in
your mind which you can use again when the need arises.”
This book grew out of various courses given at Queen’s University between 1996 and 2004. In the short span of a semester, it is difficult to cover
enough material to give students the confidence that they have mastered
some portion of the subject. Consequently, I have found that a problemsolving format is the best way to deal with this challenge. The salient
features of the theory are presented in class along with a few examples, and
then the students are expected to teach themselves the finer aspects of the
theory through worked examples.
This is a revised and expanded version of “Problems in Algebraic Number Theory” originally published by Springer-Verlag as GTM 190. The
new edition has an extra chapter on density theorems. It introduces the
reader to the magnificent interplay between algebraic methods and analytic
methods that has come to be a dominant theme of number theory.
I would like to thank Alina Cojocaru, Wentang Kuo, Yu-Ru Liu, Stephen
Miller, Kumar Murty, Yiannis Petridis and Mike Roth for their corrections
and comments on the first edition as well as their feedback on the new
material.