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.