This is a very modern text for a second course in number theory, slanted towards algebraic number theory and Diophantine equations, and using the language and concepts of abstract algebra throughout. Like most second courses in number theory it is a sampler of a wide variety of topics and does not attempt completeness in any of them. It is a translation of the 2008 French-language work Arithmétique published by Calvage et Mounet.
The book attempts, usually successfully, to cover not only modern methods but the most recent results as well. The most notable item included is probably the AKS algorithm from 2004 that shows that primality testing can be done in polynomial time. The lengthy last chapter touches on a number of topics where there have been exciting recent developments.
The exercises are especially good, and supplement the exposition with a number of important results. Many of the exercises are quite challenging, and some may have harder proofs than the things that are proved in the body.
There are several other good books for a second course in number theory. Pollack’s Not Always Buried Deep: A Second Course in Elementary Number Theory is an interesting recent book that also has very challenging exercises, but is slanted very much toward analytic number theory and so does not overlap the present book very much. Another good book that covers much the same topics as the present book from a much more traditional viewpoint (but was last revised in 1990 and so does not have the most recent topics), is Ireland & Rosen’s A Classical Introduction to Modern Number Theory.
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.