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Zbl 1155.11002
Stein, William
Elementary number theory. Primes, congruences, and secrets. A computational approach. (English)
[B] Undergraduate Texts in Mathematics. New York, NY: Springer. x, 166~p. EUR~34.95/net; SFR~58.50; \$~49.95; \sterling~27.99 (2009). ISBN 978-0-387-85524-0/hbk; ISBN 978-0-387-85525-7/ebook

Oh no, not yet another introduction to elementary number theory! Like so many other books, this one treats topics that have become standard in recent years (the six chapters cover primes, congruences, public-key cryptography, quadratic reciprocity, continued fractions and elliptic curves), and it has exercises with selected solutions. \par One cannot expect fantastic new proofs in an area which has been covered in countless textbooks, and so such a book has to be judged by its exposition. In this respect, the author succeeds admirably: the book is written in an entertaining style; it even contains a few jokes, such as Lenstra's proof that there are infinitely many composite numbers, and subjects that are usually quite dry are presented in a lively way. In fact, my favorite chapter in this book is the one on continued fractions. In (too) many sources, discussing continued fractions means going through an almost endless string of identities proved by induction. The basic relations are proved here, too, but then the text quickly moves on to interesting problems, such as Euler's expansion of $e$ or problems concerning the continued fractions of algebraic numbers of degree $>2$. \par What distinguishes this text from other books is the computational approach, which the author takes seriously. He uses the software system Sage developed by himself throughout the book, from computing greatest common divisors to proving the associativity of the group law on an elliptic curve, or for determining its rank. It doesn't take much to predict that software systems will play an ever increasing role in the future of mathematics, and having a text explaining a powerful system such as Sage has two advantages: it gives the students a tool to do calculations that illustrate even the most abstract concepts, and, simultaneously, introduces them to an open source software that can later be applied profitably for studying research problems. \par I have noticed only one error: there is a (small but serious) gap in the proof of quadratic reciprocity via Gauss sums (if $R \subset S$ are rings and if the congruence $a \equiv b \bmod q$, where $a, b, q \in R$, holds in $S$, then it does not necessarily hold in $R$; this conclusion is in general not valid if e.g. $q$ becomes a unit in $S$), and closing this gap requires a little algebraic number theory (or the introduction of finite fields). \par Yet another introduction to number theory? Yes, but an excellent one, with the additional bonus of a) presenting material that can be covered in one semester, and b) introducing the readers to a powerful software system.
[Franz Lemmermeyer (Jagstzell)]

MSC 2000:: *11-01 Textbooks (number theory)
11Axx Elementary number theory

Keywords: primes; congruences; primitive roots; Artin's conjecture; cryptography; algorithmic number theory; quadratic reciprocity; continued fractions; elliptic curves; factorization methods; primality tests

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