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Saunders Mac Lane and Garrett Birkhoff
American Mathematical Society/Chelsea
Publication Date: 
Number of Pages: 
AMS/Chelsea Publishing 330
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Michael Berg
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Now that I am of a certain age, and given that I teach at an undergraduate institution, I find myself more and more out of touch with what’s done in graduate schools in this country.  Of course, there’s the influx of recent PhD’s hired in our department with great frequency (mine is an ambitious university), but they just make me feel more out of touch.  Oh, well, tempus fugit.  So, in preparation for doing the present review, I went online to check out what texts are being used in graduate algebra at a(n algebraic?) variety of graduate schools, starting with my own, UCSD: in 2016 it was the book by Dummit and Foote.  UCLA, last year: Lang (Aha! An old favorite).  Harvard: Dummit and Foote.  OK, then, what about Yale?  It’s Lang (not surprisingly, given Lang’s affiliation).  Three schools my department sent recent graduates to:  Ohio State: Lang again.  Purdue: Dummit and Foote. UCSB: Dummit and Foote as well as Lang.  Very interesting, and, I must say, encouraging: Lang’s book was on the menu back in my day (the late 1970’s and early 1980s), so I guess I’m not that much out of the loop after all.  On the other hand, Dummit and Foote is alien to me.  This is not surprising, given that its first edition appeared in 1990 and my graduate school days were well behind me by then.
But isn’t it interesting that the styles of the books cited are so very different?  Serge Lang was notorious for his acerbity, as famously illustrated by his chapter on homological algebra in the first edition of Algebra (check it out, if you don’t know this tale), and his book is relatively austere.  For one thing, examples are not abundant, even though his coverage of algebraic structure is fabulous.  I think his coverage of Galois theory is by far the best around, at this level.  That said, UCSB probably has the right idea along these lines by requiring a supplementary text.  The point is, I suppose, that a true grounding in algebra is multifaceted and dependent to a large degree on the sensibilities of the particular graduate student.  It is on this count that I want to commend and comment on the book under review, Algebra, by Saunders Mac Lane and Garrett Birkhoff: even at this point in time, it is a wonderful entry in the race, doing things, as it were, differently than Lang --- in just the right way.   For example, granting Lang his condescension toward homological algebra, there is no denying that it, and its overarching discipline, category theory, have won the day.  Just imagine doing contemporary mathematics without a very strong grounding in this subject (cf. Charles Weibel's An Introduction to Homological Algebra.)  
One of the marvels of the book under review is its trajectory toward this goal, even while doing full justice to everything else the budding Ph.D. student needs know about algebra.  Indeed, Mac Lane – Birkhoff is one of my favorite books in this subject, pitched perfectly at the indicated level, and it sets the stage very well for, e.g., Weibel’s book, or, my personal favorite and a classic of classics, Cartan – Eilenberg, Homological Algebra.  
One small caveat is in order: “Mac Lane – Birkhoff “is not “Birkhoff – Mac Lane,” the latter being, once upon a time, the most popular book on abstract algebra at the undergraduate level.  The second book is titled, A Survey of Modern Algebra, and, while it is still a very nice way into this field for rookies, is by no means a graduate textbook.  Mac Lane – Birkhoff, Algebra, is just that, and I claim that even now, anyone studying for the Ph.D. qualifying examination, anywhere, needs to have access to it.  Saunders Mac Lane and Garrett Birkhoff were fine algebraists, masters of their craft, and pioneers to boot: Mac Lane, together with Sammy Eilenberg, could claim credit for the inception and subsequent development of category theory, and Birkhoff was a trailblazer in lattice theory.  


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


  • Sets, functions, and integers
  • Groups
  • Rings
  • Universal constructions
  • Modules
  • Vector spaces
  • Matrices
  • Special fields
  • Determinants and tensor products
  • Bilinear and quadratic forms
  • Similar matrices and finite abelian groups
  • Structure of groups
  • Galois theory
  • Lattices
  • Categories and adjoint functors
  • Multilinear algebra
  • Appendix: Affine and projective spaces
  • Bibliography
  • Index