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Elementary Number Theory: An Algebraic Approach

Ethan D. Bolker
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Luiz Henrique de Figueiredo
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In the original preface of the book, the author says that the goal of the book was to teach number theory by using the theorems of basic abstract algebra, which students sometimes find too dry when too much emphasis is placed on the axiomatics. In the preface to this Dover edition, the author suggests that both number theory and abstract algebra should be taught together, one feeding the other. This approach makes sense because the historical development of much important algebra was motivated by problems in number theory — new algebraic tools provided illuminating solutions for old problems.

The book largely achieves its proposed goals. Its main themes are the structure of the ring Zn of residue classes modulo n and its group of units, Pell's equation and its relation to quadratic fields, and Fermat's equation xn+ yn= zn. Surprisingly, Bolker does not frame the Chinese Remainder Theorem as a ring isomorphism and so cannot deduce immediately the induced isomorphism between the corresponding group of units.

The book reminded me of LeVeque's Fundamentals of Number Theory, which is in the same spirit but contains material — such as number-theoretic functions, distribution of primes, and diophantine approximation — that is omitted by Bolker because it is not algebraic in nature.

Despite the large number of misprints, which are listed in an errata at the end of the book, the book is very well written: the prose is agreeable and the exercises are fun and sometimes surprising. Even if you do not base a course on Bolker's book, it can be used as a source of ideas and examples, especially in introductory abstract algebra courses.

Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.

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