Elements of Number Theory (Undergraduate Texts in Mathematics)

Stillwell's book Elements of Number Theory presents a grand picture, starting with solving integer equations, and then working into general solutions of the Pell Equation. He covers basic ground, but without the generally random approach that most Number Theory books present. In particular, he stresses repeatedly the importance of Unique Factorization, and in many ways his book is a build up to how Unique Factorization is "saved" in general algebraic structures by Prime Ideals. That by itself makes it worthwhile, because it is bringing in very important topics from Modern Algebra that are used in modern number theory. This is especially true because that angle is missing from several introductory texts on the subject (such as Dudley's classic text). Great exercises, clear writing, and motivating discussion also gives the reader all important context. Too many number theory books show random results which, while pleasing, seem arbitrary. Here is a coherent development of deep ideas and a striking big picture. I am definitely glad to have it on my shelf.

I enjoy the algebraic approach to number theory.

After reading the first couple of chapters of Mr. Stillwell's book on number theory I've made plans to purchase his other college-level textbooks, including Elements of Algebra, Real Numbers, Four Pillars of Geometry, and, recently, Reverse Mathematics. I'm a mathematically-minded CS student and my interests fit very nicely with Mr. Stillwell's, plus I enjoy his style tremendously. Book is perfect for self-study.

This is a very pleasant introduction to number theory. Each chapter is preceded by a preview and concluded by a discussion to make the main ideas clear and well-motivated and to show how things fit in the big picture by discussing the historical development. The book starts with the very basics and moves via some pearls like the four square theorem and quadratic reciprocity to a culmination with algebraic number theory. A readable and elementary introduction to algebraic number theory is especially valuable today because, as Stillwell argues in his preface, this is the proper setting in which to learn of rings and ideals. Nowadays, of course, the custom is to pull these concepts out of a hat in the mysterious context of "abstract algebra" where there is no apparent reason to introduce them whatsoever. Fortunately, Stillwell has provided us with an equally enjoyable book on algebra, so now we can only hope that some day the curricula will change accordingly.

One of the best introductory books I've seen. It is concise, and yet very illuminating and down to earth, leading very nicely into the beginnings of algebraic number theory.

The whole book is centred on understanding vector spaces and linear mapping without recourse to determinants which are left to the end of the text. By the time you have finished the book you will certainly having a thorough grounding in this important topic. It is highly abstract but avoids the axiom, proof, theorem recipe often found in books covering Linear Algebra. The author does point out that this is written for an readership that has already met the subject in a previous course. It will certainly suite undergraduates in their second or third year or for casual readers who are more familiar with vectors in Euclidian space. Thoroughly recommended.

excellent book