Jasbir S. Chahal

Professor of Mathematics, Brigham Young University.
ISBN: 0-9740427-4-9
Kendrick Press, Inc. (2006) vi, 139 pp. $60.00.

The author presents a concise introduction to algebraic number theory suitable for a beginning graduate course. The book is based on lectures given at the University of Salzburg and at Brigham Young University. Many exercises are incorporated into the text. As for prerequisites, the reader should be familiar with linear algebra, and some basic abstract algebra including Galois theory.

The methods used are classical, along the lines of the work of Dedekind and Hilbert. The Minkowski geometry of numbers is developed and used as a tool. The author takes pains to show how commutative algebra and algebraic geometry originate in the subject matter of algebraic number theory. Students who decide to specialize in algebraic number theory will find the suggestions for further reading helpful.

Algebraic number theory is one of the great accomplishments of pure mathematics. This was the view of David Hilbert and Hermann Weyl, for example. This brief introduction conveys the power and elegance of the subject, and tempts the reader to delve further into the literature.

Chapter 1: Basic Concepts. Chapter 2: Arithmetic in Relative Extensions. Chapter 3: Geometry of Numbers. Chapter 4: Analytic Methods. Chapter 5: Arithmetic in Galois Extensions. Chapter 6: Cyclotomic Fields. Chapter 7: The Kronecker-Weber Theorem. Chapter 8: Zeta Functions and Riemann Hypothesis.