A year ago I supervised one of our seniors in a capstone seminar on Galois theory, she had just finished an REU which had turned her on to this part of algebra. She had in her possession already some specialized notes written for the REU, but she needed both the foundation of the subject and a broader (and deeper) view. I decided she should look at the excellent, if very terse, book by Emil Artin, Galois Theory. It did the job, at least to an extent: not surprisingly, she went through it very slowly (as she should), and I ended up writing a long note-set for her, which served at least some relevant purpose. Now that I am again looking at Kaplansky’s book for the purposes of this review, I wonder if I shouldn’t have gone with that source instead --- it is such a beautiful book … Well, actually, no, that would have been a bad move. It is not for undergraduates unless they’ve already had the according undergraduate course in abstract algebra, replete with a first thorough treatment of Galois theory. That, actually, was my experience back in the Jurassic Age. At UCLA in the 1970s coverage of majors’ algebra at the undergraduate level meant, in sequence, the books by Neal McCoy and I. N. Herstein (yes,

*Topics* …), and we did it in three quarters, the third covering Galois Theory. I’d done all that very early on, so in my senior year I took the graduate course and was fortunate enough to be taught by one of my very favorite professors, the late Ernst Straus. He presented us with a marvelous course, in his own style: fabulous lectures without notes, with excellent textbooks as supplements. Actually, as I recall, when he was going at group theory he stayed pretty close to Rotman's

*An Introduction to the Theory of Groups*, but he winged it much more when he was playing with rings and fields. However, he did point us in the direction of the book under review, and rightly so.

Looking at it again after all these years, I am struck by what an idiot I was not to dissect Kaplansky’s book at that time. I learned my algebra from some other great texts including the classic by Van der Waerden, Modern Algebra, and the aforementioned book by Artin. I was enamored of the Göttingen tradition, having read the book Hilbert, by Constance Reid, and went with the according sources. Well, in order to learn this great stuff as part of the regular curriculum, a baby graduate student would be better off to look at a book written for that curriculum, and then Kaplansky is pretty much unbeatable.

The book is actually a compendium of notes (very carefully written ones) for three U Chicago courses back in the 1960s, on fields, rings, and homological dimension of rings and modules. Given its origin in lecture notes, it is perhaps a bit terse (no chattiness anywhere), but, man, the proofs are nice. It is worth it to work through them just to get the feel of how a first-class algebraist does algebra. Not being an algebraist myself, but just a fellow-travelling number theorist, I was nonetheless close to many hard-core algebraists, including my dear friend and UCSD graduate school office mate, the late Erazm Behr, who had a close connection to the Chicago algebraists. So, through him, and maybe through osmosis of a sort, I became more aware of a certain style of doing this stuff: Herstein’s book, for example, mentioned above, began to take on another hue. And Kaplansky’s book fit the bill to a tee.

*Fields and Rings* is a fantastic book, perfect for self-study and graduate school, and I am going to indulge myself a bit and order an old hardback copy for a smile. I already have a couple of soft-cover versions, my own from the ‘70’s plus a gift from a retired colleague, but it’s worth dishing out another five bucks or so to get a hard-cover version, especially in this age of the machine when books are becoming all too rare.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.