"Monoids,R-Modules And Nonassociative Rings-These Are Some of My Favorite Things: A Suggested Reading List For Graduate Algebra"



Yello.

Brief political comment first.

November 2nd, 2010 may go down as one of the blackest in American history in coming decades and centuries. People who have been predicting a corporate takeover leading to a plutocracy in America -a “third world” complete with utopian gated communities defended by mercenary private security interspersed with regions of abject poverty where starving mobs of disease riddled peasants try and stay alive into their 30’s if possible-may have seen the first step towards that future on that day.


Not that our precious President and his party have been anything even remotely resembling heroic figures-between half-ass health care reform, continuing to allow the slaughter of an entire generation of young Americans, as well the slow bankruptcy of out nation to satisfy whatever mysterious powers now profit from it-they've been as cowardly and subservient to the special interests as the other side on most days. In many ways, that’s partly what led to his party’s downfall.


But with the election shenanigans with The Chamber Of Commerce-after the worst Supreme Court decision for the country at large since Plessy Vs. Ferguson - effectively rendered the Republican party a wholly owned subsidiary of the corporations. Unless a dramatic opposition to this systematic subversion of the government occurs in the next 2 years, the very best we have to look forward to from this decision is eternal gridlock until the Commerce can buy the Presidency in 2012 and put the best candidate money can buy in there.

The poor ignorant slobs who marched for the Tea Party are in for a very rude awakening indeed-that is, if any of them have the intelligence to realize they’ve been used and discarded.

And then the future I predicted in my opening preamble-coupled with environmental collapse with only the wealthy having food and water fit for human consumption, let alone other resources-will come to pass within a generation.

That’s all I wanted to say on this for now. There’ll be much more to come when I can steal myself to a full analysis and discussion.

But now onto a promised, much more academic matter.

For those who don’t know, I’m sort of an unofficial bibliophile for mathematical education. I inherited this love of textbooks and monographs from my inspiration, friend and unofficial mentor, Nick Metas. I was 18 years old when out of simple curiosity I called him in his office to ask him for direction in independent studies of mathematics beyond calculus-and he went on for 4 hours, naming just about every textbook and describing the subject of mathematics.

That long-ago conversation is what started me on the path to becoming a mathematician.

Nowadays, the influence of Nick is very clear in my life: I have an extensive library of textbooks and monographs, people ask me all the time for references on subjects and I review books for the Mathematical Association Of America’s website. I have an opinion on most commonly used texts and monographs for all subjects-and I’m reading more every year. In fact, I have a private dream of beginning my own small publishing company someday.


(Of course, that’ll depend on the prediction above being dead wrong. We can all hope the country comes to it’s senses before its too late. They’re angry-that’s a good beginning Now they just have to develop enough intelligence to channel the anger constructively. Hope it happens in time. So far it doesn‘t look promising……..)

I’ve been asked many times to compose a master list of my favorite textbooks and/or monographs. The list will probably undergo many revisions and additions before it reaches final form-but more importantly, I’ve decided to compose it in modular form i.e in components. This way, it’s broken into bite-sized components of manageable length that I can post here.

It seems to me if I wait and try to compose it all at once-well, I’ll end up writing a 2,500 page book from the old age home I’ll be dying of cancer in. So let’s get started and hope that what little insights I can give can help neophyte students looking to broaden their knowledge base in subfields of math or are just looking for a little help in coursework they’re struggling in. Comments, input and suggestions are, of course, very welcome.

The first module here is my favorite subject in all of mathematics: algebra. (A ludicrous but sadly mandatory clarification: When a mathematics student or mathematician says ‘algebra’; it’s supposed to be understood he or she means abstract algebra. High school algebra is, of course, the simplest special case of this wondrous arena. )

How do we define abstract algebra? Like most branches of modern mathematics, attempting a simple nonmathematical definition for non-mathematicians is a nearly impossible Catch-22 since it requires mathematical concepts to even attempt a meaningful definition. Entire philosophical treatises could probably be written attempting to answer the question and would probably fail. But I think we can try for a reasonable working definition here.

I think the best way to define algebra is that it is the general study of structures in mathematics. By a structure, we mean some kind of set -by which we mean naively a collection of objects-and a function f closed on S (the range of f is a subset of S) with a specified list of properties that characterizes that structure. For example, a group is a nonempty set S with a binary operation f such that f is associative, there is a unique element e in S such that for all elements a in S, f(e,a) = f(a,e)= a and for every a in S, there’s a unique a* such that f(a, a*)= f(a*,a)=e. Algebra deals specifically with these kinds of objects.

The pervasiveness of algebra in modern mathematics in the 21st century is astonishing. It’s more then the sheer scope of algebra itself, but the fact that most of the active areas of mathematics would not even exist without it.


And I’m not talking about high-tech fields where algebra’s role is obvious-like deformation theory and higher category theory. I’m referring to the fact that most areas of mathematics are formulated in the 21st century in terms of algebraic structures. To give just one possible example of a legion, modern differential geometry would be unthinkable without the language of vector spaces and R-modules. Without tangent spaces and their associated local isomorphisms, it would be impossible to generalize calculus beyond Euclidean space. It would also be impossible to precisely define differential forms, without which most of the most interesting developments of manifold theory fall to dust. As a result, a student that’s weak in algebra needs to seriously reassess a career in mathematics.

So the least I can do is give my 2 cents on the current crop of books available.

The actual direct impetus for me writing up and posting this list was Melvyn Nathanson at the City University Of New York Graduate Center. This semester, the eminent number theorist is teaching the first semester of the year-long graduate algebra sequence there. I began this semester sitting in on his lectures in order to begin preparations for the algebra half of my oral qualifiers for the Master’s Degree in pure mathematics at Queens College. Unfortunately, a combination of personal and financial issues prevented me from attending regularly. So that was the end of that.

I found Dr. Nathanson’s (he hasn’t told me it’s ok to call him Melvyn yet, so I’m going to be extra cautious as not to offend him) comments on the subject very interesting, as he has his own unique take on just about any subject. As proof, I offer this excerpt from the course’s syllabus:


In 1931, B. L. van der Waerden published the first edition of Moderne Algebra,two classic volumes, written in German, that were based in part on lectures by
Emil Artin and Emmy Noether and that became the canonical work in abstract algebra." The second edition appeared in 1937, and an English version, Modern Algebra, translated by Fred Blum and Theodore J. Benac, was published in the United States in 1949 and 1950. I and many other American mathematicians learned algebra from the original English edition of van der Waerden. It is still a great work and I strongly recommend it for intensive study. The first volume of the seventh German edition of van der Waerden is also available in English translation, but I prefer the original. Van der Waerden's algebra begins with introductions to different algebraic structures. The first seven chapters are “Numbers and Sets," “Groups," “Rings and Fields, "Polyn-omials"“Theory of Fields," Continuation of Group Theory," and The Galois Theory." As proof of van der Waerden's influence, this continues to be the starting sequence of topics in most algebra courses and most algebra books, including the contemporary classic, Serge Lang's Algebra, which I also recommend. This course is different, not just in the sequence of topics, but in its philosophy. It emphasizes themes in algebra: Divisibility, dimension, decomposition, and duality,
and the course enables algebraic understanding and technique by developing these themes. The book includes all of the theorems expected in a graduate algebra course, but in a nontraditional order. The book also includes some important topics that do not appear in van der Waerden or Lang.”

The affirmed originality of the course, I don’t doubt. I’m still hoping to obtain a complete set of handwritten notes from some of my friends in the course, which is the official text for the course. The clear implication from the preface and his subsequent remarks is that Professor Nathanson hopes to eventually expand these notes into a textbook for a graduate algebra course.

Privately, I’m hoping to work with Dr. Nathanson as a PHD student eventually and if he follows through on this, perhaps I can be involved in the book’s drafting process. But that’s for the future.

His comments got me thinking about the current state of algebra courses and the textbooks that form the basis of them. Nathanson’s experiences are not unlike those of most mathematicians of his generation: van der Waerden’s classic was the source from which he learned his algebra. Later mathematicians; particularly algebracists-such as my undergraduate algebra teacher, Kenneth Kramer-learned algebra from the earlier editions of Lang’s tome. (In fact, it was more personal for Kramer. As an honors undergraduate at Columbia in the late 1960’s, he was a student in the graduate algebra course taught by Lang himself-whose resulting lecture notes ultimately evolved into the classic text.) Most of the better universities’ graduate programs adopted Lang as the gold standard of first year graduate algebra, for better or worse, after the 1960’s. With a very few exceptions, this was the story until after the turn of the 21st century, when a host of graduate algebra texts came onto the market within a 5 year period. What was once a very sparse set of choices for this course is now a wide field of markedly diverse texts, many authored by very eminent mathematicians.

What follows is my attempt to form an amateur’s guide to these texts and my corresponding brief commentary to each. As a reviewer of textbooks, it seemed under the circumstances, that providing such a list to my erstwhile classmates in Nathanson’s course-as well as the mathematical world in general-would be a very positive undertaking. I don’t know if it would be WISE, merely positive. I must add the disclaimer that I am by no means an expert; I’m merely a serious graduate student. Therefore, this reading list must be taken with a salt lick of caution as coming from an amateur and as such, it is seriously subject to revision as my knowledge grows and my mathematical style tastes change.

A major motivation in the evaluation of each of these books has been student-friendliness. Let me clarify greatly what I mean by that. A lot of top-notch mathematicians and students have an elitist, almost snobbish reaction to a textbook when you say its’ friendly. “Oh,you mean it spoon feeds the material to the brainless monkeys that pass for mathematics majors at your pathetic university? How amusing. Here at Superior U, we use only the authentic mathematics texts. Rudin.Artin Hoffman and Kunze. Alfhors. We propagate the True Word. Math is SUPPOSED to a struggle for those truly gifted enough to be worthy of it. “

Or something equally pretentiously narcissistic.


I have a LOT to say on this and related issues-but if I started going in depth about it here, I’d write an online book here. In future installments, I’ll begin to outline them in detail.
But in plain English, this is a bunch of crap.





The reason a lot of those “classic” texts are difficult to read isn’t because their authors were first-rate mathematicians and as such, their lessons are beyond the reach of mere mortals. In a lot of cases, it was simply because most of them never really thought about teaching; of being able to organize their deep understanding of their chosen fields -and as a result, they were very poor communicators.

This lack of communication skill is reflected not only in their poor reputations as teachers, so often inversely proportional to their reps as researchers-but also in the resulting textbooks. Why don’t they? Well, again, it’s too complicated to fully go into here. But I will say that part of the reason, as any research mathematician of any prominence will tell you-is that they don’t get paid the big bucks and get the fancy titles based on how well students learn from them.

The sad part is that this myth has been perpetuated by the canonization of certain textbooks as The Books for certain classes, despite the fact that most students almost overwhelmingly despise them. And the reason why is simple: They just aren’t clear and well-organized. That makes the very act of reading them unpleasant, let alone actually learning from them. For the serious mathematics major or graduate student, this makes studying from such books virtually an act of psychic self mutilation.

To the elitists, I only have the following to say: Charles

Chapman Pugh's Real Mathematical Analysis
, Joseph

Rotman's An Introduction to Algebraic Topology
, anything by John Milnor, J.P.Serre or Jurgen Jost, Loring Tu's

An Introduction to Manifolds
, John

McCleary's A First Course in Topology: Continuity and Dimension
, George

F.Simmons' Differential Equations with Applications and Historical Notes,
Charles

Curtis' Linear Algebra: An Introductory Approach
andJohn and

Barbara Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach



I challenge them to consider any of these wonderful books to be spoon feeding students-and yet, they are eminently readable and wonderfully written books. In short, they are books students enjoy reading and therefore will not only learn from them-but will want to learn from them.

But an interesting trend has resulted from this myth. The students who are talented enough to learn from these texts who go on their careers to become mathematicians- and who care enough about teaching- recall their experiences as students. They don’t want to subject their students-or anyone's students-to the same torture. As a result, they try and write alternative books for students that do what they wish those texts had. The Computer Age has magnified this effect hundredfold as such books have become ridiculously easy to produce. As a result, we’ve gotten “backlash waves” of texts as alternatives to those classic tomes that created the large diversity of texts that currently exist in the various subfields of advanced mathematics. Where once there was a bare handful of such texts to choose from, a generation later, the “backlash” creates a myriad of them.

Some examples in the recent generations of math students will illustrate. Once, Alfhors’ ridiculously difficult Complex Analysis was the standard text in functions of a complex variable at U.S. graduate programs after the early 1960’s. There were a few alternatives available in English-such as Hille's treatise or Carathedory's-but not a lot. This lead to an explosion of complex analysis texts in the 1970’s onward: Saks/Zygmund, Rudin, Bak/Newman, Conway, Heins, Greene/ Krantz, Jones/Singerman, Gamelin,- well, that list goes on and on. A similar backlash occurred in the 1960’s and 1970’s in general topology after an entire generation had suffered through John Kelley’s General Topology wrote a legion of such texts, including the classics by Willard and Munkres.

This effect has further been enhanced by progress in those fields at the research level-which results in the presentations of the standard texts of a generation becoming outmoded. The result is the “backlash” presentations can also be “upgraded” to current language. A good example is the incorporation of category theory into advanced algebra texts post-1950’s.

I strongly believe the current large crop of graduate algebra texts is the result of a similar backlash against Lang.

I’ve gone on to some length about this because I think it’s important to keep these 2 ideas in mind- the elitist conception of Great Books and the backlash against it-when considering my readability criteria for judging such texts. So without further ado, my reading list. Enjoy.

And remember-comments and suggestions are not only welcomed, but encouraged.


Part I- Graduate Warmup: These are texts that are a little too difficult for the average undergraduate in mathematics, but aren’t quite comprehensive or rigorous enough for a strong graduate course. Of course, a lot of this is totally subjective. But it’ll make good suggestions for those struggling in graduate algebra because their backgrounds weren’t quite as strong as they thought.




Topics in Algebra, 2nd Edition by I.R.Herstein
This is the book I first learned algebra from under the sure hand of Kenneth Kramer at Queens College in his Math 337 honors course. It’s also the book that made me fall in love with the subject. Herstein’s style is concise yet awesomely clear at every step. His problem sets are legendarily difficult yet doable (mostly). If anyone asks me if they’re ready to take their algebra qualifier and how to prepare-I give them very simple advice: Get this book. If you can do 95 percent of the exercises, you’re ready for anything they throw at you. They’re that good. Warning: In true old European algebracist fashion, Herstein writes his functions in the very un-Calculus like manner on the right in composition i.e. fg= gof. This confused the author of this blog initially and no one corrected him until several weeks into the course-which lead to difficulties later on. A couple of quibbles with it-the field theory chapter is really lacking. Also. Herstein tends to present even the examples-which are considerable-in their fullest generality. This makes the book harder for the beginner then it really needs to be. For example, he gives the dihedral group of rigid motions in the plane for the general n gon where n is an integer. he could start with the n=4 case and write out the full 8 member group table for the motions of the quadrilateral and THEN generalize. Still-I fell in love with this book. The presentation is considered outdated by most mathematicians now, who prefer the more geometric approach of Artin. Still, the book will always have a special place in my heart and I recommend it wholeheartedly for the talented beginner.




Algebra 2nd Edition by Michael Artin The second edition of this book finally came out in September. For awhile, it looked like it might emerge posthumously-it was so long in gestation. But fortunately, this wasn’t the case. I haven’t read it carefully yet, but from what I’ve seen of it, it looks very similar to the first edition. As for the first edition- well, I got really mixed feelings about it. Artin’s book has many, many good qualities. It’s primary positive qualities are the heavily geometric bent and high level of presentation. The shift in emphasis from the permutation groups to matrix groups is an extremely smart one by Artin since it gives one a tool of much greater generality and simplicity while still preserving all the important properties of finite groups. (Indeed, permutations are usually explicitly represented as 2 x n matrices with integer valued bases-so the result is just a slight generalization. ) This also allows Artin to unify many different applications of algebraic structures to many different areas of mathematics-from classical geometry to Lie groups to basic topology and even some algebraic geometry (!) All through it, Artin brings an infectious love for algebra that comes through very sharply in his writing. So why the hesitant recommendation?
Because it’s easy to like the book when you’ve already learned the material from other sources.
Would even a talented beginner find the book so appealing? I don’t think so. Firstly, Artin assumes an awful lot of background in his prospective students-primarily linear algebra and basic Euclidean geometry. It might have been reasonable to assume this much background in the superhuman undergraduates at MIT in the early 1990’s, but I think that’s a stretch for most other students-even honors students. Especially nowadays. Secondly, the book is organized in a very idiosyncratic fashion that doesn’t always make sense even to people who know algebra. Nearly half the book is spent on linear algebra and group theory and rings, modules, fields are developed in a very rushed fashion. Some of these sections really needed more fleshing out. Also, a lot of the group theory chapters are confusing. His discussion of both cosets and tilings in the plane are particularly discombobulated. Lastly-his choice of topics for even good undergraduates is bizarre sometimes. He writes a chapter on group representations, but leaves tensor algebra and dual spaces “on the cutting room floor”? It’s a very strange choice. The book’s main flaw is there are too few exercises and most of these are ridiculously difficult. (In all fairness, I understand this was the main problem Artin is trying to rectify in the new edition. ) That being said, for all it’s flaws, a text of this level of daring by an expert of Artin’s stature is not to be ignored. I wouldn’t use it by itself, but I’d definitely keep a copy on my desk. Apparently,though, in the 2 decades since this book was written, Artin has rethought the course and it’s structure-it remains to be seen if the new version has gotten most of the bugs out. If it has, the book will be a must-have, hands down, for students of algebra.



A Course in Algebra by E.B.Vinberg This is very rapidly becoming my favorite reference for algebra. Translated from the Russian by Alexander Retakh, this book by one of the world’s preeminent algebracists is one of the best written, most comprehensive sources for undergraduate/graduate algebra that currently exists. Vinberg, like Artin, takes a very geometric approach to algebra and emphasizes the connections between it and other areas of mathematics. But Vinberg‘s book begins at a much more elementary level and gradually builds to a very high level indeed. It also eventually considers many topics not covered in Artin-including applications to physics such as the crystallographic groups and the role of Lie groups in differential geometry and mechanics! The most amazing thing about this book is how it manages to teach students such an enormous amount of algebra-from basic polynomial and linear algebra through Galois theory, multilinear algebra and concluding with the elements of representation theory and Lie groups, with an enormous number of examples and exercises that cannot be readily found in most other sources. All of it is done incredibly gently despite the steadily increasing sophistication of the material. The book has a very “Russian” style-by which I mean the author does not hesitate to both prove theorems and give applications to both geometry and physics (!) throughout. Those who know me personally know this is a position I am very sympathetic to-and for there to be a major recent abstract algebra text that takes this tack is very exciting to me. For anyone interested in writing a textbook on advanced mathematics, this is a terrific book to study for style. It is one of the most readable texts I have ever read. An absolutely first rate work that needs to be owned by any student learning algebra and any professor considering teaching it.








Abstract Algebra, 3rd Edition by David S Dummit and Richard M. Foote Ever seen a movie or read a book where based on your tastes, everything you think and what you see in it, you should love it-but just the opposite? You don’t like it one bit and you couldn’t explain on pain of death why? THAT’S how I feel about this book, one of the most popular and commonly used books for algebra courses-both undergraduate and graduate. It has a good, very comprehensive selection of material, good exercises and lots of nice examples for the serious student. It's also presented in a completely modern manner with commutative diagrams and lots of nice nuances.

So what’s my problem with it?

Well, first of all, it’s way too expensive. You could get both Vinberg and a used copy of Artin for the same price as this book.

Second of all-it’s pretty dry and matter-of-fact. It just doesn’t excite me about algebra the way Vinberg or Herstein do. Everything’s presented nicely and clearly-but it comes off almost like a dictionary.

Lastly-the level the book is pitched at. It has pretty comprehensive coverage of the standard topics: groups, rings, field, and modules. It also contains some topics that are better suited for graduate courses- homological algebra and group representations, for example. The big problem is the book tries to cover all these topics equally. As a result, it doesn’t succeed in developing all of them well. For example, the chapters on basic group and ring theory are excellent, but modules are presented in a very general but somewhat disjointed way.Also, the level of the book mean the more advanced topics aren't covered in enough depth for a graduate course. And to be frank, a lot of the presentation of the undergraduate material is very similar to that of Herstein- except the only about half the exercises are anywhere near as interesting as the ones there.

I think this is probably what annoys me the most about this book-it comes off as a bloated, watered down version of Herstien. I think the book would have been helped enormously by being split into 2 volumes: The first volume for undergraduate topics and the second for more advanced topics. Instead, the authors tried to cram everything into one volume that can be used for both undergraduate and graduate courses and the result is quite uneven in quality.

The verdict?

It’s probably the single best reference for undergraduate algebra that exists.As such, it's very handy for looking stuff up that you’ve forgotten or preparing for qualifying exams. But as a learning textbook,it's a mixed bag. To be honest, I'd rather use Vinberg or the first volume of Knapp as a text at this level. For its price, see if you can borrow a copy instead.


The Big Three: These are the 3 textbooks that up until about 10 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 remarks of a frequent poster at Math Overflow in this regards.

Let’s call him Mr. G.


Mr. G is a talented undergraduate at one of the more prominent universities to study mathematics in the Midwestern United States. Like the author of this blog, he also has been occasionally slammed for shooting off his big mouth on MO by the moderators.


He and I have had several heated exchanged about his Bourbaki-worship. He believes that the Bourbaki texts are sacred tomes that are the only “real” texts for mathematicians and applications are for non-mathematicans. But I’ll let his own words state his position far better then I can. Here is a recent exchange between Mr. G and 2 mathematicians who are 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: An Automathography
.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 don’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 can’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. 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 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? I’m sorry, that’s not a reasonable thing to do!

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. But homological algebra?!?

(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 then 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 problem as a significant 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 algebracist, 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 then 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 a lot of students really need to learn-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.

But I think Grillet does a considerably better job as a one volume alternative to Lang.







Basic Algebra 2nd edition Volume 1 and Volume II by Nathan Jacobson

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 edition-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 commutative and non-commutative ring and module theory as well as the representation theory of finite groups-was intended as a course for strong first year graduate 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, it’s reissue by Dover in wonderfully cheap editions is a serious cause for celebration.

(BTW-an interesting story I got from one of Jacobson's former students in the 1980's:Apparently most of Jacobson's undergraduate students struggled mightily using the first volume. Eventually,he abandoned using it as an undergraduate text and switched to Herstien's text, which the students liked much better.However, the first year graduate students and advanced research students in algebra at Yale loved both books and those were the clintele Jacobson used the texts for for the rest of his career.)


Both books are beautifully and authoritatively written in Jacobson's unique style, 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. The material in both volumes on ring and field theory-both Galois theory and general fields-are fantastic.They are concise and 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-module theory in particular suffers from this organization. Personally, I didn’t find it that big an issue with a little effort-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 make 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.



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 and 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. Knapp taught both undergraduate and graduate algebra at The State University of New York at Stonybrook for nearly 3 decades.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 beyond all expectations.

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 reviews of basic number theory and linear algebra up to an honors undergraduate course in abstract algebra (groups, rings, fields, Galois theory, multilinear algebra, module theory over commutative rings). The second corresponds to a first 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 mathematics 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 group actions on sets are stressed throughout. Also, 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.) Best of all-there are hints and solutions to ALL the exercises in the back of each volume. I would love to use this set to teach algebra one day-either as the main texts, as supplements or just references.

But if you enjoy algebra, 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 algebracist 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 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. 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. 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. 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 noncom mutative aspects first, focusing mainly on group theory. He then goes on to commutative theory-discussing ring and ideal theory, Galois theory and cyclotomy. 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. This isn't my taste in texts, I prefer the converse: Advanced books with a lot of explicit examples that leave theorem proofs for the reader.But in this case, Issacs writes so beautifully, it's really hard to complain. He also discusses some beautiful and very unusual topics, like x-groups and finite character theory. Unfortunately,for the most part,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-which is of course, a huge problem. How can you call such a book in 2010 a graduate course?It would be very hard to use such a text by itself as a graduate course in algebra-it would need to be extensively supplemented to provide modern topics any student would need for a qualifying exam. That being said-it is wonderfully written.If supplemented by a text on homological algebra, it could certainly serve as half of such a course. Either Osborne or Joseph Rotman’s books on the subject would fill in the omissions very nicely.The final chapter of Ash would also serve this need well and inexpensively.

Of course, that begs the question-if Issacs' book needs this much supplementing, then why both spend the considerable cost of the book? While I like the book immensely,I suggest borrowing a copy rather then emptying your bank account on it. A used copy for gotten online on the cheap would be a great asset, too.

And now-my very favorite algebra book of all time. Drum roll,pleeeeeeeeease................





Advanced Modern Algebra 1st edition by Joseph J. Rotman
I haven’t seen the second edition, but I’m very familiar with the first. Rotman may be the best writer of algebra textbooks alive. Hell, he may be the best writer of university-level mathematics textbooks PERIOD. Serious. 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. 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 deserves 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 proof, read the following excerpt from the first edition, how Rotman explains the basic idea of category theory and it’s 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).

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 presents algebra as a huge, beautiful puzzle of interlocking pieces-one he knows as well as anyone in the field. The one minor complaint is the book’s exercises-they’re a little soft compared to the ones in Hungerford or Lang. And the sheer size of the book-1000 hardback bound pages!- is a bit daunting. ( Rotman joked with me via email that more then a few times, he mistakenly carried it to his calculus class and had to go back to his office to switch books.) But these are very minor quibbles in a book destined to become a classic. If I had to choose one textbook for graduate algebra and it’s qualifier and couldn’t pick any others-THIS is the one I’d pick, hands down.

Word from the AMS and those who have seen it that the second edition is even better-the index has been greatly improved and entire sections have been rewritten to emphasize noncom mutative algebra-which is appropriate for a graduate course.

I suggest you all place your orders now. You’ll thank me later, I promise.

I now return you to your regularly scheduled lives.

Thank you for your attention.

Peace.


Other books mentioned in this post(worth checking out):