This is a thorough and modern introduction to elementary number theory, that introduces many advanced topics early and has excellent exercises. Despite being twenty years old, the book is more modern than most introductory text books. It uses the language of abstract algebra throughout, and has a heavy emphasis on algebraic number theory and elliptic curves. It also has quite a lot on *p*-adic numbers, which although hardly a modern subject is not usually covered in textbooks at this level.

The narrative in the book usually covers just the main outlines of the subjects, with many interesting and advanced theorems being covered in the exercises. (There are hints and answers for all exercises in the back of the book. Most of the hints are “big hints” that are really sketches of the solution.) As a result the book is less austere than some; the narrative is leisurely, but it still manages to pack a lot of content into a relatively small space by the choice of exercises.

In coverage and difficulty the book is comparable to Hardy & Wright’s An Introduction to the Theory of Numbers and to Niven & Zuckerman & Montgomery’s An Introduction to the Theory of Numbers. (Ireland & Rosen’s A Classical Introduction to Modern Number Theory has many similarities in approach to Rose, but has a substantially different coverage and is not really comparable.) Hardy & Wright has no exercises, while Rose has many. Hardy & Wright is more systematic, and therefore more useful as a reference, while Rose is more discursive and jumps around from one interesting result to another. Both books are comprehensive, although Hardy & Wright is slanted toward prime numbers and arithmetic functions and Rose is slanted toward diophantine equations. Niven & Zuckerman & Montgomery is closer to Rose in many ways, in particular in being chatty and moving a lot of material into the exercises, but like Hardy & Wright it is slanted toward the primes, does not use much of an algebraic approach, and is relatively weak on diophantine equations and has little on elliptic curves.

Bottom line: a better and more thorough introductory textbook than most, but one that requires a lot of effort and mathematical maturity from the students.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.