Algebraic Number Theory

This book offers the basics of algebraic number theory for students and others who need an introduction and do not have the time to wade through the voluminous textbooks available. It is suitable for an independent study or as a textbook for a first course on the topic.

The author presents the topic here by first offering a brief introduction of number theory and a review of the prerequisite material, then presents the basic theory of algebraic number theory. The treatment of the subject is classical as developed originally by the German school and summed up in Hilbert's Zahlbericht. Commutative algebra and algebraic geometry originate in the subject as presented here.

The newer approach provides a broader theory to include the arithmetic of algebraic curves over finite fields, and even suggests a theory for studying higher dimensional varieties over finite fields. The purpose of the last chapter is to indicate how the subject treated in this book leads naturally to the Weil conjecture and some delicate questions in algebraic geometry. In this chapter the author discusses, without supplying complete details, some advantages of this approach to algebraic number theory.

The book is inspired by famous classics as cited in the Bibliography. Even though written about a century ago, these classics are still very readable. A completed course in number theory, linear algebra, and abstract algebra should more than suffice as a prerequisite. Although all the Galois Theory needed is covered in Chapter 2, some familiarity with it will be helpful.