The first edition of Richard A. Mollin’s textbook “Algebraic number theory” [CRC Press Series on Discrete Mathematics and its Applications. Boca Raton, FL: Chapman and Hall/CRC. xiv, 483p. (1999; Zbl 0930.11001)] was published in 1999, and reviewed shortly thereafter. Now, more than a decade later, the author has brought out the second edition of this meanwhile quite popular and well-established primer, just a few months after the appearance of his most recent book “Advanced number theory with applications.” Discrete Mathematics and Its Applications. Boca Raton, FL: CRC Press. xiii, 466 p. (2010; Zbl 1200.11002)].
Actually, what one usually expects from the second edition of a well-liked, widely used textbook is the correction of former misprints and inaccuracies, occasional improvements of the presentation and possibly the appropriate adoption of some supplementary, perhaps even up-dating topics. In fact, the present second edition of Richard A. Mollin’s text “Algebraic Number Theory” is not only characterized by all those usual features, but appears to be a virtually new book. As the author points out in the preface, comments from numerous instructors and students have led him to rewrite and reorganize the material completely, thereby changing significantly both the style of exposition and the original methodology. Compared to the first edition, this already becomes apparent from the table of contents of the new edition at hand, which now reads as follows:
Chapter 1: Integral domains, ideals, and unique factorization;
Chapter 2: Field extensions (and Galois theory);
Chapter 3: Class groups (with geometry of numbers and Dirichlet’s Unit Theorem);
Chapter 4: Applications. Equations and sieves;
Chapter 5: Ideal decomposition in number fields (with Kummer extensions, class field theory, and primality testing);
Chapter 6: Reciprocity laws;
Appendix A: Abstract algebra;
Appendix B: Sequences and series;
Appendix C: The Greek alphabet;
Appendix D: Latin phrases.
More precisely, what differs from the first edition can be briefly summarized through the following facts:
(1) More basic, purely algebraic background material from the theory of commutative rings and fields, including a more elaborated treatment of Galois theory, has been brought to the main text by means of the new two first chapters.
(2) Class groups are studied much more systematically and thoroughly in the new Chapter 3, which also contains an additional treatment of binary quadratic forms as a basic methodological tool in this context.
(3) The applications of the foregoing theory to the study of special Diophantine equations, factoring algorithms, and the number field sieve have been put into a (new) separate Chapter 4, where they also have been given a partly more comprehensive treatment.
(4) Generally, in this edition, there is much less emphasis on using exercises for completing proofs of results discussed in the main text. In fact, there are now more detailed explanations of proofs, whereas exercises are referenced in the proofs only when they are of routine character and not too difficult for a novice to tackle.
(5) Any sketchy discussion of elliptic curves, as it occasionally appeared in the first edition of the book, has been completely omitted in the current second edition. As for this topic, the reader is referred to the author’s recent, more advanced book mentioned above [Advanced number theory with applications. Discrete Mathematics and Its Applications. Boca Raton, FL: CRC Press. xiii, 466 p. (2010; Zbl 1200.11002)], where a whole chapter is devoted to elliptic curves and their applications to cryptography.
Now as before, this second edition of Richard A. Mollin’s introductory textbook “Algebraic Number Theory” is an excellent source for a basic one-semester course on the senior undergraduate and beginning graduate level. The completely rewritten and reorganized version of the original is presented in a much more systematic, streamlined, comprehensive, self-contained, and didactically refined form, while all the commended peculiar features of the latter have been preserved all through. Again, each section comes with a large number of instructive examples and carefully selected exercises, where solutions to the odd-numbered ones are provided at the end of the book, and there are many mini-biographies of famous mathematicians who significantly contributed to the development of the topics treated in the text.
The author’s expository mastery, mathematical erudition, and elevated cultural taste are also revealed through the various quotations of historical celebrities, which pleasantly relax the often intimidating gravity of an abstract mathematical subject. Finally, the particular user-friendliness of this fine textbook is enhanced by both a rich bibliography referring directly to the main text, and an utmost detailed index for maximum cross-referencing.
All in all, this book is highly recommendable for anyone looking for a profound and versatile, nevertheless lucid and stirring introduction to algebraic number theory.