I'm going to soon review the 3rd edition of Saunders MacLane And Garrett Birkoff's Algebra at my blog soon and this is the first time I'm really carefully reading it. While I'm really enjoying the book,I've got a real dilemma gnawing at me. What kinds of students can this book be used for in 2017?

I understand from professors who were students at Harvard that MacLane himself would use the book-in all it's versions,from the first mimeographed drafts in the early 1960's to the 3rd edition he used in his last teaching days in the mid-1990's-for both undergraduate and graduate courses in algebra depending on the strength of the students, which would vary enormously from class to class. Usually,for strong undergraduate courses, he'd supplement Algebra with examples drawn from he and Birkoff's classic undergraduate text, A Survey Of Modern Algebra. For a first year graduate course-which at Harvard,usually included a number of undergraduates-he'd use the book straight up and cover most of it.

My question: What level students today can Algebra be used for? I think it would definitely be too difficult for any but the very best undergraduates and I think there are several books that would be better for that purpose: Herstein's Topics in Algebra or Vinberg's A Course in Algebra come to mind. I don't think hitting students with category theory unless they've already had considerable exposure to algebra is a good idea. I doubt they'll be able to fully appreciate the enormous generalization and unification it provides without a giant stock of examples (which is why I love Emily Riehl's Category Theory In Context as the definitive introduction to category theory for mathematics students. She gets this.) But at the same time, since the book contains very little homological algebra and no representation theory beyond the definition,can it really be used as graduate course today?

Would like to hear feedback on this question.


1 Answer 1


Preface. Birkhoff & Mac Lane's Algebra is a brilliant book. I should probably spend some time with it again, actually. Also, I apologize for such a long response. I think too much about algebra pedagogy and textbooks. The short version is I think the book can be used for either undergraduates or graduates with some success, but I think it is less than ideal in both cases.

Preliminary Thoughts

In both undergraduate courses and graduate courses students' abilities and background are quite variable; that much has not changed in the past few decades. There are still many overqualified students and many underqualified students, and I can provide a constructive proof of that fact. Students of the first type would benefit from a more challenging text, whereas students of the second type would benefit from an easier one.

I do not believe students are significantly less capable today than they were several decades ago, as you seem to suggest. In fact, they may be better when it comes to algebra. Algebra was almost unheard of at the undergraduate level not too long ago, whereas now an undergraduate program would be remiss not to require students take at least an introductory course covering the basic theories of groups, rings, and fields. On the other hand, calculus has been pushed into high school, and introductory courses are substantially easier and less rigorous than in the days of Apostol, Courant, and Spivak. In this way, I do not expect today's students are substantially better than past students, either. Also keep in mind that Dummit & Foote is fairly commonly used at the undergraduate level, and not just at top schools or in honors classes. I would not consider that presentation substantially easier than Birkhoff & Mac Lane, albeit perhaps a bit.

Potential Courses

I think the text could be used for either undergraduate or graduate students with decent results, but I do not think it is the best choice as a primary text. Anyway, let me say a bit about my thinking for how this book would work for either audience.


Birkhoff & Mac Lane is ill-suited for standard courses at schools with typical undergraduates; however, schools where the average undergraduate is exceptional could probably make do and similarly for honors classes at other schools.

As Mac Lane did years ago, it is best to supplement the text with something easier, something intended for undergraduates. You mention Herstein and Vinberg, which are good; Artin or Gallian also work, for example. (I have come to like Gallian quite a bit since first giving it a real chance to see what my school teaches.)

It is unlikely you will cover the entire book, either, even in a one year course. A typical first undergraduate course may cover group theory through the isomorphism theorems and the structure theorem for finite abelian groups,possibly including group actions and the Sylow theorems (c.f. Mac Lane Chapters I, II, VII), basic ring theory (c.f. Chapter III), and some field and vector space theory (c.f. Chapters VI, VII). As aforementioned, some category theory, especially an awareness of universal properties and basic definitions (e.g., categories, functors), can be good, and it seems to be becoming more common, too (c.f. Chapter IV). A second course may cover a bit more group theory (c.f. Chapter VII), more field theory and linear algebra (c.f. Chapters IX-XI), and Galois theory (c.f. Chapter XIII).

Mac Lane has an enduring reputation as a great expositor. Indeed, I find his writing quite agreeable. That the basic objects of groups and rings are presented early on and come back to several times, rather than spending one-hundred pages only talking about groups, then changing to only talking about rings, should also help students see how algebraic structures are related; it also makes the beginning easier, and provides an opportunity for students to save themselves if they fall behind early on.

The biggest problem is that Birkhoff & Mac Lane use categorical algebra all from Chapter III onward, which reflects the fact that they intended readers to be fairly mature. Although not enough to reconcile this problem, their exposition on categorical notions is quite clear, at least, albeit maybe not as good as some treatments that have since come. They do not avoid using universal properties, and they do not always bother to give students something concrete to hold on. Young students can handle abstraction. Some people mistakenly believe they cannot. Being able to handle abstraction does not mean students should not learn many concrete, basic examples, however, nor does it mean they should learn things at the most abstract possible level and be expected to figure out the less abstract consequences on their own. Maximum generality entailing sophisticated machinery can seem efficient in the abstract, but it rarely works so well in practice. You do not define a group as a groupoid with one object, at least not in an introductory course. (This is not to say Mac Lane and Birkhoff do this!)

This is one of the primary reasons an undergraduate course ought to supplement this book. In fact, I think this book works better as the supplement.

The truth, though, is that undergraduates are fairly unlikely to read their textbook. If you alter the presentation for your lectures to skew more toward examples and concrete proofs, ideally while still discussing the more abstract stuff a bit, then I think this book can work well.

Or, maybe you have a truly prodigious bunch who have all seen algebra before or who are as mature as most incoming graduate students at good schools, in which case following the text more closely ough to work without much issue.


Again, I think Gallian's Contemporary Abstract Algebra is better as a primary text. Motivation, examples, clear writing, reasonable exercises, they are all there. Artin (2e, not 1e) is also good and emphasizes linear algebra and geometric intuition, which is good considering how often students will need things from linear algebra and how often they will find themselves ignorant of those things. Vinberg seems similar but more intense, so I imagine it would work well, too. Dummit & Foote is not good enough for ring theory and too encyclopedic to be used as the primary text for an undergraduate course, in my opinion; the group theory is so good, though. Aluffi is really something special, but it is probably a bit too category theory-centric for students new to the subject in the sense that they will not have enough familiarity with the lower-level stuff. Judson is good all around. If you want an older book written by a master expositor and mathematician, then I think Herstein works better than Mac Lane. It has a lot of linear algebra, which is good, and it is not too hard, but it requires some work. It also introduces the student to modules, but it does not insist on working with modules instead of vector spaces whenever possible, which is probably good, because modules often serve to slightly confuse without adding anything more than a bit of generality.

Some combination of the approaches of the above books ought to make an excellent undergraduate course.


I agree that having no real homological algebra or representation theory is a major drawback of Birkhoff & Mac Lane. But, hey, at least there are lattices! (I am joking, if it is not clear.) I also think there should probably be more commutative algebra. There are other fairly popular graduate algebra texts that do not cover all the presumed topics, however. Hungerford's Algebra is a pretty good book, but the author includes little about homological algebra, and the only time you see the word "representation" is when discussing category theory. Rotman's Advanced Modern Algebra is pretty light on the Galois theory (but he gets points for having a chapter on algebras). Aluffi (Is Chapter 0 an undergraduate or graduate text, or is it neither?) also fails to include much Galois theory while completely ignoring commutative algebra and representation theory (but he gets points for recognizing theses omissions and planning a Chapter 1). It is also worth noting Mac Lane does cover multilinear algebra pretty well, which is frequently forgotten.

Perhaps not every course or book needs to cover all of these topics in detail, though, as much as the books by Lang and Dummit & Foote would have one believe. I am not sure. Certainly, algebraists expect everyone has seen some homological algebra and representation theory by the time they get their Ph.D., but plenty of people do not need either, both topics are often taught in separate courses, and so forth.

I think you could cover the entire book, minus perhaps Chapter XIV on lattices, which are not typically emphasized and are partly historical, and possibly Chapter VIII, which is a bit odd, in my opinion. Supplementing this with another book to get some coverage of representation theory and homological algebra is probably ideal. Dummit & Foote or Lang are the obvious places to go, but I do not think those books' best parts cover these topics. Unfortunately, I am not sure what the best supplements are. I quite like Etingof's Introduction to representation theory, which is available at his site, but it takes two chapters before it gets to the case of finite groups, which is usually what algebra professors focus on in a first graduate course. I do not think using algebras is a problem, but three whole chapters is probably a bit much, so some would have to be cut. I think taking parts from Rotman's An Introduction to Homological Algebra may work in a similar vein, albeit with much more cutting. The advantage of not falling back on a general reference as a supplement is you can use the really great parts of these excellent books, and the students know where to go for much more; plus, both of these are written at a reasonably low level. Adding in some applications may be good, too; I do not remember many being in Mac Lane. Robert Ash's Algebra: The Basic Graduate Year includes some good applications to algebraic geometry and algebraic number theory, for example.


I am less sure what makes a really excellent graduate course in terms of extant texts. I love Lang, especially for things like Galois theory, but it is too hard, too fast, too big, too encyclopedic, and, dare I say, too modern for most graduate students. Certainly, one can survive it, but it is probably suboptimal for most. Lots of people cannot stand Lang's writing, too. Dummit and Foote is great for group theory, but it suffers in other parts and may also too big and encyclopedic. I often found it dry, too. Similarly, books like Cohn, Grillet, and Jacobson can be too advanced or too focused on being references. Rowen has been talked about a good bit, which is deserving for its extensive presentation on algebras and many applications, but I am not sure starting with modules is a good idea, for example.

I think graduate courses should use category theory pretty openly. This is when students should come face-to-face with having to understand universality, or else. Yet, I also think a course should start with basic material. Perhaps a lecture reviewing elementary set theoretical notions, then cover some linear algebra (introducing basic category theory after seeing direct sums), then cover some ring theory, then plenty of group theory, then modules and advanced linear algebra, followed by field and Galois theory, representation theory (using algebras and specializing quickly to groups), commutative algebra (including some applications to algebraic geometry and the like), and finally homological algebra, with some advanced or extra topics at the end, if possible (e.g., a word on universal algebra or further category theory). Linear algebra is first, because students have the best intuition there; ring theory is next, because the examples and applications are nicer there than in groups and the quotient construction is easier. Logic suggests the standard groups, rings, fields, modules, vector spaces, etc. sequence, but it does not demand it.

The idea is to a.) cover all of the absolute basics of graduate algebra (e.g., basic modules, Jordan-Hölder), b.) cover many of the further topics (e.g., inverse limits, infinite Galois theory), c.) get some glimpses into truly modern topics, even research (e.g., deformation rings), and d.) see some applications (e.g., the fundamental group, valuations, schemes). Because there is quite a bit, and because students are assumed to be mature and to be getting used to reading math, not just going to lecture, one can easily assign boring lemmas and the like as exercises or reading (but only truly easy ones as exercises!). The lecture ought to focus on the main ideas, examples, and results, and the book ought to clearly show what is important, very important, etc.

Getting across the rich motivation, ideas, analogies, history, and so forth should be an explicit goal, but it will not always be possible. Ideally, the course book would have lots of this and the lecture somewhat less. There should also be specific portions of book and lecture that are intentionally plain, definition-theorem-proof type things, but where all the richness is introduced later down the line, as to not snub the students. This will get everyone to do the work they will need to do to read many papers, which are not uniformly well written and often take having a mature audience as an excuse not to motivate or show any work.

On the other hand, there should be parts where the students are discovering things or work from ideas, trying to work out the formality. Developing the ability to take some vague notion and work out the details is key to graduate education. Again, eventually the students should be given this stuff to them. We must not risk them never learning it.

I think these principles in lecture and text, along with hard work by the students, lots of student-student communication (including some forced working together), some student-professor communication (including some forced communication, e.g., an oral exam or a special, discussion-based lecture), good homework (comprising most of the grade and including a good number of routine exercises, perhaps even a few utterly obvious ones, a few intermediate ones, and a few truly hard ones, perhaps in many steps to resemble research or requiring some additional reading), and reasonable exams would make for a nearly perfect course. Theoretically, less-prepared students and more-prepared students would not only survive but also get from this something substantial. I think the needs of the graduate student are more complex, in a sense, than the undergraduate's needs, because many are less obvious than in the undergraduate case. For undergraduates, we obviously need to ensure they see examples, learn the basic theory, be able to write proofs, etc., and those are hard things to do, but it is easier to recognize what those things are and there are fewer of them than for graduate students.

Where it Belongs

This is one of many books I believe is best used as a supplement or for review/reference. At both the graduate and undergraduate level, it seems there are books that are better in parts or in whole. This book also does not match a clear (unorthodox) vision of how the subject should be taught, unlike, say, Aluffi or Adkins. Those are perhaps worth substantial time, even if they are somewhat too odd for most courses. It is a good, once standard book, but there are texts that better suit the needs of modern students.

  • $\begingroup$ A wonderful and detailed response. I've already commented extensively on most of the standard algebra texts at my blog-you can find the relevant post here: tableschairsandbeermugsmathemagician.blogspot.com/2015/08/… The only missing major texts I didn't comment on are MacLane/Birkoff,Weintraub/Adkins (which a lot of people don't know about,surprisingly) and Rowen and I hope to do so in future posts. $\endgroup$ Commented Sep 6, 2017 at 18:41
  • 1
    $\begingroup$ (cont) Rowen is basically an updated version of Jacobson at a slightly lower level. The most impressive thing about Rowen is how up to date it is.It contains many topics you won't find in any other standard graduate algebra book, such as an extensive section on nonassociative algebras and Zelmanov's theorem on Jordan algebras. Also,the author writes very well indeed. Because of it's unorthodox structure and old fashioned style,though,I'm not sure I'd use the book for a graduate course. But I'd sure keep one on reserve in the library for my students to browse. $\endgroup$ Commented Sep 6, 2017 at 18:46
  • $\begingroup$ I agree, I do not believe Rowen is all that fitting for a standard (or nearly standard) course. It is a great supplementary text, however. I am hard pressed to put a single book above the others, however. I mentioned flaws of a few books in my answer. Hungerford is a common alternative I did not discuss, but I have never found it very exciting. Rotman may be a good primary text. I think using many texts is often (and in this case almost certainly) a good idea. Supplementing Rotman with Aluffi, D&F, and Lang, for example, may work well. $\endgroup$ Commented Sep 6, 2017 at 19:01
  • $\begingroup$ Also, your review page is quite comprehensive! I feel there is a need in mathematics for detailed discussion of available resources. We have websites like Math Stack Exchange where we can ask users for opinions, but it would be nice to have many more bibliographies of mathematical subjects than there currently are. Surely, people appreciate the extant ones (e.g., the famous Chicago bibliography). I once began writing a massive annotated bibliography of basic mathematics (through standard graduate work), but after spending ten pages on algebra texts, I gave up on the project. $\endgroup$ Commented Sep 6, 2017 at 19:28
  • 2
    $\begingroup$ You should have never given up on the project. I think such a book would have found a massive audience and I'd encourage ANYONE working on such a book to continue as we'd then have many different opinions for students and professionals on standard works. I'm working on such a book myself,but for low cost texts only. It's a projected book to accompany my new publishing company to instruct penniless students on low-cost mathematics sources for self study. Sadly,in America,such a book will be VERY important soon for the serfs. $\endgroup$ Commented Sep 6, 2017 at 19:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.