Graduate Texts in Mathematics. 211. New York, NY: Springer. xv, 914 p. EUR 74.95/net; sFr. 124.50; £52.50; $ 69.95 (2002).
The first edition of this book (1965; Zbl 0193.34701) consisted of three parts: basic concepts, field theory and linear algebra, and as a modern down-to-earth approach with a personal touch it attained great popularity. The second edition (1984; Zbl 0712.00001) added some topics, mainly on commutative algebra and homological algebra.
The current third edition has grown again, the additions dealing with topics close to the author’s heart from number theory, function theory and algebraic geometry. For the math graduate who wants to broaden his education this is an excellent account; apart from standard topics it picks out many items from other fields: Bernoulli numbers, Fermat’s last theorem for polynomials, the Gelfond-Schneider theorem and (as an exercise, with a hint) the Iss’sa-Hironaka theorem. This makes it a fascinating book to read, but despite its length it leaves large parts of algebra untouched. Semisimple algebras get a very cursory treatment (no mention of crossed products or the Brauer group) and there is only the merest trace of Morita theory; there are no Ore domains, Goldie theory or PI-theory. Graphs, linear programming and codes, constructions like ultraproducts and Boolean algebras are also absent, and lattices are only of the number-theoretic sort (reseaux, not treillis).
Bearing these limitations in mind, the reader will nevertheless find a very readable treatment of many modern mainline topics as well as some interesting out-of-the-way items.
Editorial comment: Note that there is also a 3rd ed. published by Addison-Wesley 1993 reviewed in Zbl 0848.13001.