Customer Reviews

Introductory Algebraic Number Theory

5 star:		(1)
4 star:		(0)
3 star:		(1)
2 star:		(0)
1 star:		(0)

 

 

Strengths:

1. Easy reading, detailed proofs

2. Covered required algebra background (modules, ideals, Dedekind domains, etc)

3. Many, many examples

Weaknesses:

1. Too detailed in some cases

2. Does not develop more advanced ideas that actually make the material easier

3. Poor index

4. Examples are often too simple

This book takes the reader through the required algebra background and moves them into the realm of using these abstract algebraic construction to study the theory of numbers. The book is aimed at upper-level undergraduates, so it's easy reading. Sometimes too easy reading, as proofs are often long-winded and contain many trivial details. In some instances, I wanted all those details, often it was simply annoying.

The real strength of this book lies in the many explicit examples. It was worth the price for these examples, as most higher-level books offer few examples.

The index is terrible, but the additional reading section at the end of each chapter is a nice addition.

Overall, I learned a lot from this book, but would have liked to have the authors approached the material at a little bit higher level. For instance, instead of using complex conjugates extensively, I would have preferred introducing a mapping to the complex conjugates (say sigma) for use in most proofs.

There are only 2 genuine introductory texts to algebraic number theory---this book and the one by Stewart and Tall. The latter is not as inclusive as the present text. This text abounds in examples. Unlike the other reviewer, I do not find them tedious, but explicit instead. Both Williams and Saban are specialists in cubic equations, and the text is interestingly flavored with this expertise. There is a very detailed, and theoretic, introduction to Minkowski bounds, class group numbers, units of general number fields, and factoring in a tower of domains. As an amateur mathematician, I am grateful to both authors for setting down their insights in a readable and graspable manner. They invite the reader to accompany them on an exciting journey into a beautiful realm of mathematics. This text will enable the reader to tackle, later on, a more formidable book like "Algebraic Number Theory" by Mollin. For example, a problem found on page 10 of Mollin's book is found on page 136 of this text. If one has any hope to master Mollin's deep meditation on ideals, one would first need to become fluent in Williams' and Saban's presentation of them.