**The Big Three:**These are the 3 textbooks that up until about 15 years ago, were the standard texts at the top graduate programs in the U.S. to use for first year graduate algebra courses and for qualifying exams at PHD programs in algebra. Of course, at such programs, the line between graduate and undergraduate coursework is somewhat ambiguous. But I think most mathematicians would agree with me on this assessment.

*Algebra 3rd edition* by Serge Lang Ok, let’s get the elephant in the room out of the way first.

Lang is a good example of the kind of strange “canonization’ of textbooks in academia which I’ve mentioned before at this blog and other places. It’s funny how some mathematicians-particularly algebracists at the more prestigious programs- that get very self-righteous and uppity when you question whether or not Lang should be used as a first-year graduate text anymore with all the new choices. I can’t help but use some of the past remarks of once-frequent poster at Math Overflow in this regards. Let’s call him Mr. G.

Mr. G was a talented undergraduate at one of the more prominent universities to study mathematics in the Midwestern United States. ( I have no doubt that since that exchange, Mr. G has gone on to bigger and better things-assuming nothing tragic has occurred to block his way-and is working on his PhD in mathematics at a prestigious university. I also have no doubt he treats most of his students like Neanderthals with Down Syndrome and uses Baby Rudin to teach calculus. ) Like the author of this blog, he also was occasionally slammed for shooting off his big mouth on MO by the moderators.He and I had several heated exchanged about his Bourbaki-worship: G believed that the Bourbaki texts are sacred tomes that are the only “real” texts for mathematicians and applications are for nonmathematicans. But I’ll let his own words state his position far better than I can. I have in mind a specific exchange that took place in 2010 between Mr. G and 2 mathematicians who were also at the time frequent posters at MO: let’s call them Dr. H and Dr. L. This was a question regarding the presentation of graduate algebra. (I obviously can’t be more specific then that-to do so would identify the participants.)*@Dr. H: The first graduate algebra course is often going to be the student's first introduction to algebra. It's supposed to be abstract and intense! If you muddy the waters with applications, your students will never get to that level of Zen you achieve after stumbling around in an algebra course. It's like point-set topology, except the rabbit-hole called algebra goes much deeper and is much more important. –Mr.G@Mr.G: After my first algebra course I still didn't understand why I should actually much about Galois theory from a practical point of view until I saw $GF(2^n)$ in all sorts of applications. My experience has been that most students--even graduate students studying algebra--are not going to be interested in abstraction for its own sake. Mechanics can help to motivate calculus. The same can be true of information theory and algebra. –Dr.H@Dr.H: Graduate math students shouldn't be taught things "from a practical point of view". This isn't a gen. ed. class, and the abstract perspective one gains by really engaging algebra "as it is practiced) is completely worth the "journey in the desert", as it were. This is the "Zen" I was talking about. Also, I think that characterizing algebra as "abstraction for abstraction's sake" is really missing the point tremendously. – Mr.G.@Mr.G. The journey through algebra does not necessarily have to go through the desert, nor is that necessarily the best or most ideal path. It might be so for you, but it is certainly not the best path for everybody. There are numerous other paths to take, most of which can lead and have lead people to mathematical understanding and success. Please take a moment to consider, for instance, Richard Borcherds' recent algebraic geometry examples post. – Dr. L.*

Speaking for myself, I firmly believe in heeding Lebesgue’s warning about the state of the art in mathematics: “ Reduced to general theories, mathematics would become a beautiful form without content: It would quickly die.” Generality in mathematics is certainly important, but it can and often is, overly done. But I digress. My point is that Mr.G’s attitude is typical of the Lang-worshipper: That if you can’t deal with Lang, you’re not good enough to be a graduate student in mathematics. Or to use Mr.G’s own words on another thread on the teaching of graduate algebra: ”Lang or bust.” Many feel the “journey in the desert” of Lang is a rite of passage for graduate students, much as Walter Rudin’s

*Principles of Mathematical Analysis*is for undergraduates.

Well, there’s no denying Lang’s book is one of a kind and it’s very good in many respects. People ask me a lot how I feel about Lang’s remarkable career as a textbook author. It’s important to note I never met the man, sadly-and everything I know about him is second hand.

Reading Lang’s books brings to mind a quote from Paul Halmos in his classic autobiography I Want To Be A Mathematician .which was said in reference to another famous Hungarian mathematician, Paul Erdos: “I don’t like the kind of arithmetic-geometric-combinatorial problems Erdos likes, but he’s so good at them, you can’t help but be impressed. “

I didn’t like the kind of ultra-abstract, application-devoid, Bourbakian, minimalist presentations Lang was famous for-but he was SO good at writing them, you couldn’t help but be impressed.

And

*Algebra*is his tour de force.

The sheer scope of the book is stunning. The book more or less covers everything that’s covered in the later editions of van der Waerden-all from a completely categorical, commutative-diagram-with-functors point of view. There’s also a generous helping of algebraic number theory and algebraic geometry from this point of view as well, much more so then any other texts at this level. His proofs are incredibly concise and with zero fat, but quite clear if you take the effort to follow them and fill in the blanks. The chapters on groups and fields are particularly good. Lang also is an amazingly thorough and responsible scholar; each chapter is brimming with references to original proofs and their source papers. This is a book by one of the giants in the field and it’s clearly a field he had enough respect for to know his way around the literature remarkably well-and he believed in giving credit where credit is due. Quite a few results and proofs-such as localization of rings, applications of representation theory to functional analysis and the homology of derivations, simply don’t appear in other texts. The last point is one I think Lang doesn’t get a lot of credit for, without which the book would be all but unreadable: He gives many, many examples for each concept-many nonstandard and very difficult to ferret out of the literature. Frankly, the book would be worth having just for this reason alone.

So fine, why not go with the party line then of “Lang or bust”?

Because the book is absurdly difficult, that’s why.

First of all, it’s ridiculously terse. It takes 2-3 pages of scrap paper sometimes to fill in the details in Lang’s proofs. Imagine doing that for OVER 911 PAGES. And worse, the terseness increases as one progresses in the book. For the easier topics, like basic group theory and Galois theory, it’s not so bad. But the final sections on homological algebra and free resolutions are almost unreadable. You actually get exhausted working through them.

AND WE HAVEN’T EVEN TALKED ABOUT THE EXERCISES YET.

“Yeah, I’ve heard the horror stories about Lang’s exercises in the grad algebra book. C’mon, they’re not THAT bad are they?”

You’re right, they’re not.

They’re WORSE.

I mean, it’s just ludicrous how hard some of these exercises are.

I’ll just describe 2 of the more ridiculous exercises and it pretty much will give you an idea what I’m talking about. Exercise 30 on page 256 asks for the solution of an unsolved conjecture by Emil Artin. That’s right, you read correctly. Lang puts in parentheses before it: “The solution to the following exercise is not known.” No shit? And you expect first year graduate students-even at Yale-to have a chance solving it? I’m sorry, that’s not a reasonable thing to do! (One does wonder if a brilliant student in one of Lang’s courses had actually succeeded in solving the problem whether or not they’d get credit for it. Or if suddenly a few months later there’d be a new interview with Lang in the New York Times science section about the brilliant Lang’s new discovery solving Artin’s conjecture while the student scrambles to pay his or her student bills while eating ramen noodles. We’ll never know-thankfully.)

Then there’s the famous-or more accurately, infamous-exercise in the chapter on homological algebra: “Take any textbook on homological algebra and try and prove all the results without looking up the proofs.”

I know in principle, that’s what we’re all supposed to do with any mathematical subject we’re learning.

(An aside: I actually had a rather spirited discussion via email with Joseph Rotman, Professor Emeritus of the University of Illinois at Urbana-Champaign, over this matter-a guy that knows a thing or 3 about algebra. Rotman felt I was too hard on Lang for assigning this problem. He thought Lang was trying to make a point with the exercise, namely that homological algebra just looks harder than any other subject, it really isn’t. Well, firstly, that’s a debatable point Lang was trying to make if so. Secondly, I seriously doubt any graduate student who’s given this as part of his or her final grade is going to be as understanding as Rotman was. Actually, it’s kind of ironic Rotman thinks that since I know many a graduate student who would have failed the homology part of their Lang-based algebra course without Rotman’s book on the subject! )

My point is I don’t care how good your students are, it’s educational malpractice to assign problems like that for mandatory credit. And even if you don’t and just leave them as challenges for the best students-isn’t that rubbing salt in the wounds inflicted by this already Draconian textbook?

These exercises are why so many mathematicians have bitter memories of Lang from their student days.

A lot of you may be whining now that I just don’t like hard books. That’s just not true. Herstien is plenty difficult for any student and it’s one of my favorites. In fact, quite the contrary. You’re really supposed to labor over good mathematics texts anyway-math isn’t supposed to be EASY. An easy math textbook is like a workout where you’re not even winded at the end-it’s doubtful you’re going to get any benefits from it.

But Lang isn’t just hard; I don’t just mean students have to labor over the sections before getting them.

The average graduate student learning algebra from Lang’s book is like a fat guy trying to get in shape by undergoing a 3 month U.S. Marine Corp boot camp and having a steady diet of nothing but vitamins, rice cakes and water. Assuming he doesn’t drop dead of a heart attack halfway through, such a regimen will certainly have the desired effect-but it will be inhumanely arduous and unpleasant.

And there are far less Draconian methods of obtaining the same results.

So unless one is a masochist, why in God’s name would you use Lang for a first year graduate course in algebra?

Is it a TERRIBLE book? No-as I said above, it has many good qualities and sections. As a reference for all the algebra one will need in graduate school unless becoming an algebraicist, the book is second to none.

Would I use it as a text for a first year graduate course or qualifying exam in algebra?

HELL NO.

** Algebra by Thomas Hungerford** : This has become a favorite of a lot of graduate students for their algebra courses and it’s pretty easy to see why-at least at first glance. It’s nearly as demanding as Lang-but it’s much shorter and more selective, has a lot more examples of elementary difficulty and the exercises are tough but manageable.

The main problem with this book occurs in the chapter on rings and it boils down to a simple choice. Hungerford-for some strange reason-decides to define rings without a multiplicative identity.

I know in some ring theoretic cases, this is quite useful. But for most of the important results in basic ring and module theory, this results in proofs that are much more complicated since this condition needs to be “compensated” for by considering left and right R-modules as separate cases. Hungerford could alleviate this considerably by giving complete, if concise, proofs as Lang does in most cases.

But he doesn’t. He only sketches proofs in more than half the cases.

The result is that every section on rings and modules is very confusing. In particular, the parts on modules over commutative rings and homological algebra-which students really need for qualifying exams-are all over the place.

And the field theory chapter is a train wreck, frankly.

Still, the book has a lot of really nicely presented material from a totally modern, categorical point of view. The first chapter on category theory is probably the best short introduction there is in the textbook literature and the section on group theory is very nice indeed. So I ultimately have mixed feelings about Hungerford. I’d definitely give it a look and if I could borrow a copy, great. But I’m really not sure I’d want to shell out money for my own copy. I’d rather get a copy of Grillet or Rotman.

** Basic Algebra, 2nd edition by Nathan Jacobson, volumes I and II:** I say we should nominate Dover Books for a Nobel Peace Prize for their recent reissue of this classic. The late Nathan Jacobson, of course, was one of the giants of non-commutative ring theory in the 20th century.

He was also a remarkable teacher with an awesome record of producing PhDs at Yale, including Charles Curtis, Kevin Mc Crimmon, Louis H.Rowen, George Seligman, David Saltman and Jerome Katz. His lectures at Yale on abstract algebra were world famous and had 2 incarnations in book form: The first, the 3 volume Lectures In Abstract Algebra, was for a generation the main competition for van der Waerden as the text for graduate algebra courses. Basic Algebra is the second major incarnation- the first edition came out in the 1970’s and was intended as an upgraded course in algebra for the extremely strong mathematics students entering Yale from high school during the Space Age. The first volume-covering classical topics like groups, rings, modules, fields and geometric constructions-was intended as a challenging undergraduate course for such students. The second volume-covering an overview of categorical and homological algebra as well as the state-of-the-art (circa 1985) of non-commutative ring theory-was intended as a graduate course for first year students. The complete collapse of the American educational system in the 1990’s has rendered both volumes useless as anything but graduate algebra texts. Still, given that the second volume was going for nearly 400 dollars at one point online in good condition, its reissue by Dover in wonderfully cheap editions is a serious cause for celebration.

Both books are beautifully and authoritatively written with a lot of material that isn’t easily found in other sources, such as sections on non-associative rings , Jordan and Lie algebras, metric vector spaces and an integrated introduction to both universal algebra and category theory. They **are** rather sparse in examples compared with other books, but the examples they **do** have are very well chosen and described thoroughly. There are also many fascinating, detailed historical notes introducing each chapter, particularly in the first volume.

The main problem with both books is that Jacobson’s program here absolutely splits in half algebra into undergraduate and graduate level topics; i.e. without and with categorical and homological structures. This leads to several topics being presented in a somewhat disjointed and inefficient manner because Jacobson refuses to combine them in a modern presentation. It also results in a quantum leap in abstraction between the presentations of the same material in the 2 volumes. This is really where the lack of examples in the first volume hurts the total presentation-it would be doable but very tough for even strong students to use that first volume by itself as an adequate foundation for Volume II. Module theory in particular suffers from this organization. R-modules are first presented classically in volume I over a principal ideal domains (PIDs). In Volume II, a gargantuan step in generality is taken by discussing the categories of left and right modules over arbitrary noncommutative rings. This is a bit like making the first half of an analysis course a pencil pushing calculus course with virtually no theory and then beginning with “Adult Rudin” in the second half. Ok, I’m exaggerating-it’s not quite **that** bad. But the leap in difficulty **is** pretty significant.

**and providing you use the 2 books simultaneously as a single text, that’s the key**. But a lot of other students have complained about it. Also, some of the exercises are quite difficult, rivaling Lang’s. Even so, the sheer richness of these books makes them true classics. If graduate students are willing to work a little to unify the various pieces of the vast puzzle that Jacobson presents here with astonishing clarity, he or she will be greatly rewarded by a master’s presentation and depth of understanding.

*Algebra Volumes I- III Second Edition by Paul M. Cohn*

*Classic Algebra by Paul M Cohn*

*Basic Algebra: Groups, Rings And Fields by Paul M.Cohn***Ever since I posted the first version of this list at this blog, I’ve gotten comment after comment wondering how I could have left the most famous algebra text in Europe out. To be perfectly honest, I simply hadn’t carefully read Cohn’s books back then. I hadn’t seen them except in passing in the St. John’s University library and they were far too expensive for me to purchase. In preparing this revision, I realized it would be malpractice for me to not comment on them since they’re so widely used and respected. Yes, I’m well aware the books have since been republished in new editions by Springer-Verlag as 3 separate books with the titles above. This is why I’m considering both the latest edition of the old “unified” text and the new versions simultaneously. Essentially, the new books are corrected and radically reorganized versions of the old 3 volume text.**

*Further Algebra And Applications by Paul M. Cohn :**Classic*is a new version of volume I with some additions from volume II, such as group representations and tensor products.

*Basic*and

*Further*are similarly corrected and rearranged versions of volumes II and III. Since the reissued editions don’t differ significantly in subject matter, merely organization, I want to focus on the second edition of the original 3 volumes as they’ve traditionally been used and revered in the U.K. The book is quite different from standard American algebra texts- which is why I’m going to go into some depth on the background of the book. I think this is important to fully understand the book and its’ intention.

*{Algebra}’s changing role is reflected in the importance of algebra in the curricula, as well as in the many excellent textbooks that now exist. Most of these are designed for undergraduates at North American universities and are either (a) a very broad introduction to linear algebra, with a little groups and rings, for general students taking mathematics, or (b) a course for graduates, or junior—senior students majoring in mathematics, who have already taken a course of type (a). The pattern in Britain is a little different: the honours student specializing in mathematics takes algebra for two or three years (depending on his ultimate interests) and his need is for a textbook which combines (a) and (b) above and is somewhere between them in level. The object of the present work is to provide such a book: the present first volume includes most of the algebra taught in the first two years to undergraduates at British universities; this will be followed by a second volume covering the third year (and some graduate) topics.*

**required**of many programs for

**high school students**there planning on applying to university to be taking calculus and linear algebra before graduating. The other major difference that’s relevant to the background of Cohn is that practical applications are generally sneered upon. This is not because British academics don’t think it’s important, it’s that they don’t believe a university is the place for it. A university is a place for strictly formal training in pure theory-as much as you can cram into three years of coursework. The thinking is that only in a strictly academic setting can theory truly be learned in the depth and breadth needed for professional work. Once students have that background, they’re more then capable of picking practical stuff up after graduating. This is one of the reasons they’re able to effectively condense a Master’s degree level of training into 3 years while American students take twice that-it’s much more focused and rigidly structured then in the U.S.(There’s also the completely insane manner British universities grade students, which would give me a nervous breakdown. But that’s not really relevant here, so I’ll save it for another day.)

*honors*-England and America, 2 countries separated by a common language) are even better. Freshman honours students at top universities there would probably have a good working knowledge of US-“honors” level single variable calculus and linear algebra as well as basic proof-based mathematics. In other words, they would have the equivalent mathematical background of average third year undergraduate mathematics majors in the U.S. Also, their classes would generally be very abstract with little or no applications.

*efficient.*It’s all there, but the author works hard to ensure it’s presented with the absolute minimum of verbiage. Interestingly, while there are not many

*explicit*examples-3 or 4 at most in the first 2 texts and less in the last-there are many examples embedded in the discussions of theorem proofs. Also, the explicit examples are very well chosen. So the books are actually quite bit deeper than a skimming read would indicate. There are also many exercises that are quite diverse in both type and level. Cohn writes extremely well and he demonstrates enormous command of the material. However, his style is very dry and dictatorial. If you’re looking to be inspired, look elsewhere. Personally, I’d rather use either Rotman or Rowen for a first graduate course. That being said, there’s a school of thought in mathematics that prefers texts that aren’t “too wordy”. This is the school of thought that thinks there’s no other real analysis text except “baby Rudin”. For them, I think they’ll find this book very much to their liking for both undergraduate and graduate algebra courses. It’ll also be of great use for students preparing for qualifying exams in algebra and need to review and/or learn most of the landscape of modern algebra relatively quickly and actively. It’ll also make quite a good graduate algebra text if used in conjunction with a source that supplies examples, such as Ash. As I said, it wouldn’t be my first choice for an advanced algebra text. But I

**am**happy I have a copy.

**The New Kids On The Block:**As I said earlier, the adoption of Lang worldwide as the canonical graduate algebra text had a backlash effect that’s been felt with a slew of new graduate texts. I haven’t seen them all, but I’ve seen quite a few. Here’s my commentary on the ones I’m most familiar with.

** Basic Algebra/Advanced Algebra by Anthony W. Knapp:** This is probably the single most complete reference for abstract algebra that currently exists. It is also paradoxically, the single most beautiful, comprehensive textbook on it. This is the book Dummit and Foote should have been instead of trying to cram a whole graduate course on the back end without category theory. Knapp taught both undergraduate and graduate algebra at SUNY Stonybrook for nearly 3 decades-and these volumes are the finished product of the tons of lecture notes that resulted. The purpose of these books, according to Knapp, is to provide the basis for all the algebra a mathematician needs to know to be able to attend a conference on algebra and understand it. If so, he’s succeeded

**very**well indeed. The main themes of both books are group theory and linear algebra (construed generally i.e. module theory and tensor algebra) . The first volume corresponds roughly to what could possibly be covered at the undergraduate level from basic number theory and linear algebra through all the standard undergraduate topics up to the beginning of a first year graduate course in abstract algebra (groups, rings, fields, Galois theory, multilinear algebra, module theory over commutative rings). The second corresponds to a first and second year graduate syllabus focusing on topics in noncommutative rings, algebraic number theory and algebraic geometry( adeles and ideles, homological algebra, Wedderburn-Artin ring theory, schemes and varieties, Grobner bases, etc.) This is the dream of what an advanced textbook should be-beautifully written, completely modern and loaded with both examples and challenging exercises that are both creative and not too difficult. In fact, the exercises are really extensions of the text where many topics and applications are in fact derived-such as Jordan algebras, Fourier analysis and Haar groups, Grothendieck groups and schemes, computer algebra and much, much more. The concept of group actions on sets is stressed throughout. Both books are completely modern in presentation regardless of level. Categorical arguments are given implicitly before categories are explicitly covered by giving many commutative diagram arguments as universal properties. (This avoids the trap Jacobson fell into. Knapp in many ways is a much more detailed version of Cohn’s opus.) The books are also supremely organized with each chapter independent to some degree from the others. This means the books are extremely versatile and many different kinds of courses can be based on them. Best of all-there are hints and solutions to

**all**the exercises in the back of each volume. My one complaint with the book is that several topics I feel are too important to be shortchanged are. The most glaring example is group representation and character theory. But this is really a minor quibble. In a book like this, the author has to make choices and such choices are never going to do everything perfectly in a way that makes everyone completely happy. I would

**love**to use this set to teach algebra courses one day at any level-either as the main texts, as supplements or just references. If you enjoy algebra, are learning it or plan to teach it, you

**have**to have a copy. Hopefully, there will be many editions to come.

** Abstract Algebra, 2nd edition by Pierre Grillet:** The first edition of this book was simply called “Algebra” and it came out in 1999. To me, this book is what Lang should be. Grillet is an algebraist and award-winning teacher at Tulane University. Interestingly, he apparently carried out the revision in the aftermath of Katrina. The book covers all the standard and more modern topics for a graduate course in a concise, very modern manner-much like Lang. Unlike Lang, though, Grillet is extremely readable, selective in his content and highly structured with many digressions and historical notes. The sheer depth of the book is amazing. Unlike Lang, which focuses entirely on what-or Hungerford, which explains a great deal but also is very terse- Grillet focuses mainly on why things are defined this way in algebra and how the myriad results are interconnected. It also has the best one chapter introduction to category theory and universal algebra I’ve ever seen-and it occurs in Chapter 17 after the previous 16 chapters where commutative diagrams are constructed on virtually every other page. So by the time the student gets to category theory, he or she has already worked a great deal with the concepts implicitly in the previous chapters. This is very typical of the presentation. Also, Grillet doesn’t overload the book with certain topics and give the short shriff to others-many texts are half group theory and half everything else, for instance. Grillet gives relatively short chapters on very specific topics- which makes the book very easy to absorb. The exercises run the gamut from routine calculations to proofs of major theorems. Unfortunately, the book has 2 flaws that annoy me. First of all, like Cohn and Jacobson, it has relatively few examples. That being said, it

**is**a graduate course and the examples the author

**does**choose are very good ones and not so standard. For example, both semigroups are discussed at some length and this is the only text to my knowledge that mentions the Light Cayley table associativity test. More serious is the omission of a chapter on homological algebra, which is kind of strange given the length at which category theory is discussed. The resulting text is a clinic in how to write “Bourbaki” style texts and it would be a great alternative to either Lang or Hungerford.

** Algebra: A Graduate Course by I.Martin Issacs: **This is a strange book and I have many mixed feelings about it. After being out of print for over a decade, it was recently reissued by the AMS. Isaacs claims this course was inspired by his teacher at Harvard, Lynn Loomis, whose first-year graduate course Issacs took in 1960 there. Like Loomis’ course, Issacs emphasizes noncommutative aspects first, focusing mainly on group theory and representation theory. He then goes on to commutative theory-discussing ring and ideal theory, Galois theory ,cyclotomy and many other topics. The book is one of the most beautifully written texts I’ve ever seen, with most theorems proved and most example constructions left as exercises. Unfortunately, the book is not without serious flaws. First of all, Issacs’ material choice seems to follow his memories of his graduate course in 1960 far

**too**closely-this choice of topics would be a first year graduate course at a top university

**only**before the 1960’s. Issacs omits completely multilinear algebra, category theory and homological algebra. How can you call such a book in 2015 a graduate course? More seriously, the book has virtually no concrete examples in it. Almost none. Just a smattering in the discussion and exercises. I mean it practically makes Cohn look like it’s brimming with examples. For any book written at a lesser level, that would be the kiss of death for me. But Issacs is

**so**wonderfully and deeply written-it reads almost like a novel, with so many wonderful insights. To be able to use the book for a graduate course, one would have to supplement it extensively. The problem is that Issacs is so expensive- even now in the new AMS reissue of it-that I’m not sure if it’s worth it. This is why I was hoping the book would be reissued by Dover, but it didn’t happen. Because of the cost and the strange layout of the book, I’d be really hard pressed to assign the book to my students as the text for either an honors algebra or graduate algebra text. That being said-I

**certainly**would put a copy on reserve for my students to browse.

** Advanced Modern Algebra by Joseph J. Rotman:** Rotman may be the best writer of algebra textbooks alive. Hell, he may be the best writer of university-level mathematics textbooks

**period.**(Well, ok, I wouldn’t go

**that**far. Tom Korner, John Stillwell, the late George Simmons, Peter Cameron, Gilbert Strang and several others probably beat Rotman for that title. But solely for algebra texts, Rotman is probably king.) So when his graduate textbook came out, I begged, borrowed and cajoled until I could buy it. And it was one of the best textbooks I ever bought. I’m talking about the first edition, but second edition is very similar and in many respects, even better. The contents of the book are, as the AMS’ blurb discusses:

*“This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic K-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization.”*

That says what’s in the book. What it doesn’t tell you is what makes this incredible book so special and why it deserved a second edition so quickly with the AMS: Rotman’s gifted style as a teacher, lecturer and writer. The book is completely modern, amazingly thorough and contains discussions of deep algebraic matters completely unmatched in clarity. As one simple example, read the following excerpt of how Rotman explains the basic idea of category theory and its’ importance in algebra:

*Imagine a set theory whose primitive terms, instead of set and element, are set and function.*

How could we define bijection, cartesian product, union, and intersection? Category theory

will force us to think in this way. Now categories are the context for discussing general

properties of systems such as groups, rings, vector spaces, modules, sets, and topological

spaces, in tandem with their respective transformations: homomorphisms, functions, and

continuous maps. There are two basic reasons for studying categories: The first is that they

are needed to define functors and natural transformations (which we will do in the next

sections); the other is that categories will force us to regard a module, for example, not in

isolation, but in a context serving to relate it to all other modules (for example, we will

define certain modules as solutions to universal mapping problems).

How could we define bijection, cartesian product, union, and intersection? Category theory

will force us to think in this way. Now categories are the context for discussing general

properties of systems such as groups, rings, vector spaces, modules, sets, and topological

spaces, in tandem with their respective transformations: homomorphisms, functions, and

continuous maps. There are two basic reasons for studying categories: The first is that they

are needed to define functors and natural transformations (which we will do in the next

sections); the other is that categories will force us to regard a module, for example, not in

isolation, but in a context serving to relate it to all other modules (for example, we will

define certain modules as solutions to universal mapping problems).

I dare you to find a description of category theory that would serve a novice better. The book is filled with passages like that-as well as hundreds of commutative diagrams, examples, calculations and proofs of astounding completeness and clarity. Rotman’s book is often compared with Dummit/Foote, which is one of its’ main competitors as a graduate algebra text. It really isn’t a fair comparison in my opinion. Dummit/ Foote begins from jump and presents things in greater generality. For example. D/F introduces noncommutative rings with the definition of rings. Rotman develops commutative rings in detail first. I think this approach, while more tedious, prepares a student better for a fuller discussion of both noncommutative algebra and

*Basic Abstract Algebra for Undergraduate And Graduate Students*by Robert B. Ash Ok, very quietly, this has become students’ secret weapon for their first year qualifying exams in algebra. It’s also the book that made the author nearly a household name in mathematics textbooks and it’s not hard to see why. The history of the book is remarkable-the book began life as an extensive set of lecture notes Ash began compiling nearly 20 years ago for the graduate algebra sequence at the University of Illinois at Urbana-Champaign when he was frustrated looking for a textbook his students could actually read and understand. He’d used Lang, Hungerford and Jacobson for the course with mixed results. His students really hated reading and working through any of them-especially Lang. Ash liked Lang’s choice of material and thought with a little more concrete discussion and explanation, it could be a terrific book. So he began writing a kind of “supplemental guide” to Lang with many more examples and a lot of rigorous, but concrete discussion. Over the next 2 decades, the notes evolved into the online version of the book and students all over the internet found it to be

**very**illuminating-especially when used in combination with Lang’s text. Gradually, there grew enormous demand for a printed version of the notes and Ash was able to get Dover to put out a nice, cheap paperback and the rest is history. It’s not hard to see why this book in short order has become so beloved by struggling graduate students. The book has huge coverage of the subject and a plethora of examples in each section. (Indeed, there is a supplementary section to the first 4 chapters that allows an instructor to use this part of the book as a stand alone undergraduate text!) Groups, rings, fields, modules, algebraic geometry and algebraic number theory(!) and much more. The book is incredibly clear, comprehensive and illuminating with a style that is resolutely concrete but never lacking in either precision or rigor. The real apex of the book is a nearly 60 page concluding chapter on basic homological algebra that is one of the most pellucid and detailed presentations of the subject there is. Frankly, if this was all there was in the book, it’d still be worth having. But there is so much more good stuff in here. This includes brief sidebars into major topics like p-adic analysis and group representation theory, complete with references for further study- and Ash’s book contains complete, meticulous solutions to all the exercises. There’s simply no more any student could ask for in either a course text or for self-study. This book is a true classic in every sense of the word and Ash has given a huge boon to students of all levels and generations by making it available so widely, easily and cheaply. My one quibble with the book is I wish he’d included a more detailed section on group representation theory. That subject still doesn’t have a good introductory book in the same price range. Tragically, there will be no future editions. Ash died in April 2015 when he went for his usual afternoon walk and was plowed over by a car. A senseless death in an equally senseless world. He left behind this wonderful text as part of his legacy and due to its low price, it’s guaranteed to be one of the go-to texts for serious students of algebra for generations to come.

*Algebra*by Larry Grove: For over a generation in the U.S., this was the most popular alternative for a graduate class in algebra to either Lang or Hungerford unless you wanted to use one of the old classics like van der Waerden . ( In Europe, there were a few other options, but most of them weren’t in English except for the early editions of Cohn’s books.) The best description of the book’s contents is from Grove’s own preface to the book:

*My own practice in teaching has been to treat the material in Chapters I-V (Groups, Rings, Fields And Galois Theory, Modules, Structure of Rings and Algebras ) as the basic course, and to include material from Chapter VI (Infinite Abelian Groups, P6lya- Redfield Enumeration, Integral Dependence and Dedekind Domains, Transcendental Field Extensions Valuations and padic Numbers, Real Fields and Sturm’s Theorem, Representations and Characters of Finite Groups Some Galois Groups) as time permits. There are in Chapters I-V, however, several sections that can be omitted with little consequence for later chapters; examples include the sections on generators and relations, on norms and traces, and on tensor products. The selection of “further topics” in Chapter VI is naturally somewhat arbitrary. Everyone, myself included, will find unfortunate omissions, and further topics will no doubt be inserted by many who use the book. The topics in Chapter VI are more or less independent of one another, but they tend to draw freely on the first five chapters.*

*Algebra Chapter 0*by Paulo Aluffi : Most of the books listed so far fit into one of 2 general types: They’re either gigantic texts of exhaustive detail in one or multiple volumes that can’t possibly be entirely covered in a single year from cover to cover or they’re terse, slender volumes that leave the bulk of the proofs to the students. What students really want and need is a single volume book of manageable length and depth that contains all the essential topics of modern algebra and can actually be read from cover to cover and mostly mastered in a year. It would be much easier for student and teacher alike to base a one year graduate course on such a text. We finally have such a text in Aluffi. Several things make Aluffi’s text special and different among this whole gaggle of texts. First of all, it’s completely modern at a level somewhat lower than usual. It’s pitched at a slightly higher level than Dummit/Foote, but not quite as difficult as Lang. It’s about the same level as Rotman, but is nowhere near as exhaustive in coverage. It covers all the standard topics for a graduate algebra course: groups, rings, fields, R-modules, linear and multilinear algebra and a very lengthy chapter on homological algebra. It also covers some not so standard topics, such as homotopical algebra, derived functors and group cohomology. Categories and commutative diagrams are introduced along with set theory

**at the very beginning**and in a very gentle way. He doesn’t just dump the whole functor-object-composition machine on the student’s heads and then just start running diagram chases all over the place like Lang or Cohn. He introduces the material slowly and with many examples-and most importantly, he only introduces the categorical machinery as needed. For example, although categories and commutative diagrams are both used from the very beginning, for the most of the text, he uses only standard functions in a naïve set-theory sense with formulas alongside the diagram chases. The key concept that’s used and stressed is universal properties. Functors are formally introduced much later when discussing tensor products and multilinear algebra. In many ways, Aluffi’s book is a more advanced, streamlined and better organized version of D/F. There are quite a few examples, but some sections have more than others. There are also many, many exercises, through which both the material in the book and additional main topics are developed. They range in diversity and level from additional examples to side lemmas to full blown developments of topics like the modular group, quaternions, nilpotent groups, Artinian rings, the Jacobson radical, localization, Lagrange's theorem on four squares and a lot more. The student really needs to work through as many of these as possible to get the full value of the book. The real strength of the book is the very tight unification through both category theory and the enormous number of exercises that are challenging, but not too difficult. The one omission that annoys me yet again is representation theory. In all fairness, though-it’s such a vast subject it’s very hard to construct a single chapter or set of notes that does it justice in a first year graduate course. Still, it’s an outstanding text, one of the very best I’ve seen in the batch-and it’s relative brevity and superior organization makes it much more ideal for a real world classroom setting then most of the others. If I had to choose one textbook for graduate algebra and its qualifying exams and couldn’t pick any others-

**this**is the one I’d pick, hands down. As much as I love both Rotman and Knapp, Aluffi’s book would much more practical for a student.

*Godzilla*film.

Thank you for your attention.

Peace.

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