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A Graduate Course in Algebra, Volume 1

Ioannis Farmakis and Martin Moskowitz
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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With well over thirty years having gone by since I took my PhD, I’m sure that the list of usual suspects for standard texts for the first two years of a doctoral program has undergone a lot of changes. In my day, analysis meant Royden or (green) Rudin; complex analysis meant Ahlfors or Hille (or even green Rudin again), or maybe Levinson-Redheffer; topology meant Munkres and Spanier; and algebra meant Lang. I was a contrarian, I’m afraid, and learnt most of my algebra from a book that had been the gold standard for an earlier time, van der Waerden. But I did do a lot with Lang’s book, too, as well as others such as Emil Artin’s gem on Galois Theory and Atiyah-MacDonald’s Introduction to Commutative Algebra. Yes, long ago, if not so far away.

That said, I guess it’s still fair to say that the stuff mathematical yout’s need to know before they start wading in the deeper and colder waters of thesis research is still pretty much the same. Or is it? I would guess, for example, that with the explosions in low-dimensional topology and differential geometry that occurred over the last few decades, there might be an according shift in preparation. In topology (and algebraic geometry), for example, recent developments might ask the rookies to gain familiarity with a good deal more homological algebra. And maybe this sort of rolling with the punches of the Zeitgeist is present in other areas, too.

But surely graduate algebra is graduate algebra. Well, let’s see, using the book under review as our test-case. In their Volume 1, the authors, Farmakis and Moskowitz, start with an Introduction dealing with set theory, including, meritoriously, I think, a section on transfinite induction. Next they travel a familiar route: groups, “further topics” on groups, vector spaces, inner product spaces, rings and fields and algebras, and finally R-modules. At first glance, it all looks pretty standard, even to a dinosaur like me. But there’s more here than meets the eye. For example, F & M add coverage of injective groups, Prüfer groups, Poincaré’s upper half plane, Grassman and flag varieties, and stuff on Lie groups including the Iwasawa decomposition. (I’d like to see the qualifying examination that included questions on flag varieties, actually. But it is all very good stuff.)

In Volume 2 we get more ambitious, and I note that certain topics do appear to be justified on account of relatively modern mathematical evolutions, about which I’ll say a little more presently. In any case, in the second volume we start, very properly, with multilinear algebra (including some Hodge theory!), then proceed to symplectic geometry, commutative rings with unity (F & M say “identity”), p-adic numbers, Galois theory (a little late in the game, at least according to my sensibilities, but all right), and then representation theory and associative algebras. This is all excellent: as I indicated above, in the Jurassic age, when I did my quals, something like commutative algebra needed to be outsourced, so to speak (witness Atiyah-Mac Donald, above), and non-archimedean rings and fields (positive characteristic) were by no means covered in depth in the regular course of things. It’s also very nice to see representation theory given so much airplay.

My guess is that Farmakis and Moskowitz go symplectic (unheard of in my day) because of the presence of symplectic geometry in the interface between low dimensional topology and quantum field theory, or, more prosaically, quantum mechanics itself, and then symplectic geometry with all its contemporary activity. And including representation theory with bells on is certainly proper on any number of (modern) counts. In other words, the selection of topics in this two-volume Graduate Course in Algebra testifies to the fact that mathematics, always very much alive, has undergone all sorts of growth in the last thirty years (using me as a measure of time), and this must be reflected in how fledgling graduate students are taught and prepared. This is a very positive aspect of the book under review.

As regards the nuts and bolts of this offering, I think the proofs are solid and well-done, in that the level of preparation of the reader is not overestimated. These are novices after all, and F & M know it: the book is exceptionally readable, and will build enthusiasm as well as confidence in the kid who takes it seriously. This kid should do homework, obviously, and there are lots of exercises available. And they’re good; you won’t find anything like Lang’s notorious exercise at the close of (the first edition of) his chapter on homological algebra (cf. p. 105 of loc. cit.: “Take any book on homological algebra, and prove all the theorems without looking at the proofs in that book…”).

I also like the authors’ taste in footnotes, what with their frequent emphasis on history, i.e. the minutiae of the lives of many mathematicians appearing in these pages. There remarks add a particular dimension of fun and pleasure to what I think is a very good book. It’s pitched at the right level, it does a lot of serious stuff in preparation for what is coming the students’ way in the future, and it does it well.

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

  • Introduction and Fundamentals
  • Groups
  • Further Topics in Group Theory
  • Vector Spaces
  • Inner Product Spaces
  • Rings, Fields and Algebras
  • R-Modules