The Fundamental Theorem of Arithmetic says that every integer factors uniquely as a product of prime numbers. This is a beautiful theorem that we all learn in elementary school, at which point we assume it is the end of the story. For most people it is the end of the story, and it is only those of us who are lucky enough to take some number theory in college who start to learn that there are some problems with this statement: the definition of prime numbers is actually more subtle than we were led to believe, and to truly state uniqueness carefully we must get into some pedantic points about reordering and whether negative numbers can be prime or not.
The proof of this theorem is not horrendous, but it is in no sense trivial. More importantly, if we want to do the mathematician thing of generalizing this result to other settings, it may or may not hold. In fact, factorization turns out to be a topic so rich that there are still many open questions being actively worked on, and one could write a whole book growing out of the topic.
Someone has written such a book. Steven H Weintraub has a new book entitled Factorization: Unique and Otherwise which was published this summer by AK Peters. Weintraub's book begins by analyzing factorization in some concrete analogues of the normal integers, such as the Gaussian integers and quadratic fields, while also proving more general theorems about principal ideal domains and unique factorization domains and their differences. These discussions lead to deeper looks at other topics, such as Pell's Equation, Dedekind domains, and Dirichlet's Unit Theorem.
Throughout the book, the exposition is crisp and self-contained, and Weintraub manages to strike a very nice balance between explicit computations and abstract theory. There are a good number of exercises, and plenty of directions one could go in after reading this book.
That said, this reviewer is not sure how one could best use this book. The author suggests that one possible use for this book is as a textbook to an elementary number theory course which would cover a different set of topics than the typical course in this area. I am not convinced that Weintraub has written the right book for that audience, as a reader needs to wade through some algebraic background such as integral domains and quadratic extensions before getting to the "sexier" applications which one might want to come early in a first course, and the theory would be much easier to motivate and digest for someone who has had a course in abstract algebra already.
On the other hand, the book might be too elementary for an advanced student wishing to learn more number theory: in particular, Weintraub's book is similar to Number Fields by Daniel Marcus, in that it gives some peeks into Algebraic Number Theory without all of the technical baggage that one needs to really get to the meat of that area, but Marcus' book is at a slightly higher (although still advanced undergraduate) level, even if it is starting to feel out of date.
With that hesitation in mind, I still recommend picking up Weintraub's book and hoping you can find a good use for it: it approaches elementary number theory, a topic on which hundreds of books have been written, from a new direction. For that alone it should be rewarded, and this book has far more to offer.
Darren Glass is an Assistant Professor of Mathematics at Gettysburg College whose research interests include Number Theory, Algebraic Geometry, and Cryptography. He can be reached at email@example.com