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The Theory of Algebraic Numbers

Harry Pollard and Harold G. Diamond
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Darren Glass
, on

The MathSciNet review of the first edition of Harry Pollard’s 1950 book The Theory of Algebraic Numbers was already quite positive, describing it as “A clear and rigorous exposition of the elementary part of the theory of algebraic numbers, following classical lines (as by Hilbert), but in more accessible form and with modifications of some particular points.” By the time the second edition was published in 1974, the book had undergone some revisions and improvements, including the introduction of many exercises for the reader to work out as well as a new coauthor in Harold Diamond.

Pollard and Diamond’s book introduces many of the key ideas in Algebraic Number Theory in as elementary a way as one can imagine. The book opens up with chapters on divisibility, Gaussian primes, and algebraic numbers and integers, not even introducing the concepts of ideals until the sixth chapter. At this point, the authors go on to discuss unique factorization, ideal classes, and class numbers. A final chapter gives some applications of these ideas to studying Fermat’s Conjecture.

As one might expect from these topics and the age of the book, at times Pollard and Diamond’s book feels somewhat outdated. Most of the ideas in this area of mathematics have been greatly refined over the last half century (not to mention that Fermat’s Conjecture has been proven), and a student wishing to use these ideas to do research in mathematics will need to learn a more modern point of view — and there are many great books of that type to choose from! However, this book remains a very readable introduction to an interesting area of mathematics, and many of the exercises are illuminating to think about. One could easily imagine handing this book to an advanced undergraduate student and letting them work through it — something which I am not sure I would say about all algebraic number theory textbooks. Most importantly, with a Dover edition retailing for ten dollars, it is hard to argue with adding this book to your collection.

Darren Glass is an Associate Professor and Chair of Mathematics at Gettysburg College. He can be reached at

Chapter I. Divisibility
1. Uniqueness of factorization
2. A general problem
3. The Gaussian integers
Chapter II. The Gaussian Primes
1. Rational and Gaussian primes
2. Congruences
3. Determination of the Gaussian primes
4. Fermat's theorem for Gaussian primes
Chapter III. Polynomials over a field
1. The ring of polynomials
2. The Eisenstein irreducibility criterion
3. Symmetric polynomials
Chapter IV. Algebraic Number Fields
1. Numbers algebraic over a field
2. Extensions of a field
3. Algebraic and transcendental numbers
Chapter V. Bases
1. Bases and finite extensions
2. Properties of finite extensions
3. Conjugates and discriminants
4. The cyclotomic field
Chapter VI. Algebraic Integers and Integral Bases
1. Algebraic integers
2. The integers in a quadratic field
3. Integral bases
4. Examples of integral bases
Chapter VII. Arithmetic in Algebraic Number Fields
1. Units and primes
2. Units in a quadratic field
3. The uniqueness of factorization
4. Ideals in an algebraic number field
Chapter VIII. The Fundamental Theorem of Ideal Theory
1. Basic properties of ideals
2. The classical proof of the unique factorization theorem
3. The modern proof
Chapter IX. Consequences of the Fundamental Theorem
1. The highest common factor of two ideals
2. Unique factorization of integers
3. The problem of ramification
4. Congruences and norms
5. Further properties of norms
Chapter X. Ideal Classes and Class Numbers
1. Ideal classes
2. Class numbers
Chapter XI. The Fermat Conjecture
1. Pythagorean triples
2. The Fermat conjecture
3. Units in cyclotomic fields
4. Kummer's theorem
References; List of symbols; Index