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Rant to me about algebra books *October 25, 2012*

*Posted by David Speyer in Uncategorized.*

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Next term, I will be teaching the second semester of graduate algebra here at Michigan. The big mandatory topics are finite groups and Galois theory. There is usually time for a bit more of whatever the instructor wants to fit in. I want to do some representation theory. In my dreams, we’ll also do a bit of playing with number fields, but that might be overly ambitious.

My project for the next few weekends is to skim through as many algebra texts as I can and pick one to use. So I thought I’d put up a request for your opinions. Below the fold, some of my criteria:

The text should cover finite group theory, rep theory, Galois theory and, ideally, some Dedekind domains. Abstract linear algebra, including tensor products, would also be a strong plus, although in theory they’ve all had that already.

I’m a dynamic lecturer who is good at generating excitement and drawing connections. (Or so I like to tell myself.) By comparison, I am not as good at presenting technical arguments and definitions. I believe that teachers should choose textbooks which complement their style, so I would prefer a book which is careful and precise at the expense of being duller.

I’d like a good reference book. These are grad students, or undergrads who are very likely to go to grad school. The textbook should be useful to them beyond the class.

Ideally, I’d like a book which shows off connections of algebra to the rest of mathematics. All of our grad students take this course (except for those who already know the material), including lots of analysts and geometers. Let’s convince them algebra is useful and beautiful.

In recent years, the course has been taught from Dummit and Foote, from Artin, and from Lang. I definitely plan to look at these. My favorite algebra text is Jacobson, but I think I have to reject it on the grounds that he doesn’t do rep theory until the middle of volume 2, after a lot of other intimidating stuff. (I love this book, though, so feel free to talk me into using it.) Please let me know other great options I’m missing, or what you think of these.

## Comments

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“careful and precise at the expense of being duller”

I think you can forget about Lang then. That book has all: typos (in exercises, particularly), subtle flaws in proofs, bad writing etc.. The good part about Lang is the choice of topics. If you are willing to write lecture notes that completely replace the book, then Lang is a good start, but I would never throw the book itself at students.

I don’t have much to tell about the other texts, except that they’re probably much better. A problem with Dummit and Foote is that they treat representation theory merely as a tool for studying finite groups, which is not very motivating to people like me (compared with a modern treatise like in Etingof’s lecture notes). At a cursory look, they seem to do Galois theory right (fundamental theorem proven without primitive elements — check).

Artin’s representation theory chapter seems tailored for physicists; there is not much algebra going on there (or maybe I’ve seen it so often that I don’t notice it anymore).

I know this is stupid, but it seems to me like the book that best fits your description — minus the applications — is “Algebra” by Jantzen and Schwermer, unfortunately in German…

IIRC good things have been said about Falko Lorenz’s “Algebra I and II”. As you can see from the table of contents, the choice of topics is highly nonstandard, but nothing keeps you from throwing away the ones you don’t want. Representation theory is done from a modern perspective, apparently. I don’t have any personal experience with the book, though.

You might want to have a look at Rotman’s “Advanced Modern Algebra” published by the AMS, 2010.

A relatively new book that takes a very modern and interesting approach is Aluffi’s “Algebra: Chapter 0″ (AMS, 2009). It starts by introducing just enough category theory (essentially the definition, properties of morphisms and universal properties) and only then dives into groups. We are using it in a master’s level algebra course at Penn.

I would normally recommend Aluffi, but it’s mostly perpendicular to David’s choice of topics (no representation theory, not enough Galois theory).

While I was writing a “Guide” to algebra for the MAA’s series, I had a chance to look at a lot of these books. I think for the course you outline I would go with Rotman. The only downside of his Advanced Modern Algebra is sheer size: it’s a true monster of a book that would have been more useful split in two. But it’s elegantly written and friendly, and has everything you need in it.

I agree with Darij on Lang: very good taste, but not really executed well. Rowen’s Graduate Algebra seems too advanced for the course you describe. I didn’t look too carefully at Grillet’s book, but the table of contents looks very good.

In the end, however, my conclusion was that the best books are the ones that treat a single subject, not the encyclopedic reference books. But you probably know that, and it doesn’t solve the problem of choosing a textbook!

I was taught out of Dummit and Foote. It does finite group theory very well, and goes a brilliant job at field theory and Galois theory. (The exercises are great.)

The module theory section is not very good. The tensor product portion gives little motivation for the construction.

But, overall, I really like the book. It seems to be the right level for undergraduate and masters students taking their first or second semester graduate algebra course.

Milne has a set of notes on group theory / rep theory of finite groups and a set of notes on Galois theory. They are freely available on his website http://www.jmilne.org/math/

Since you asked us to rant at you, how about you skip all the stuff which none of your students will ever use in their lives. The key example of which is, of course, the Sylow theorems.

Just out of curiosity I’ve searched for recent texts published in Algebra. Turns out all those that appear more relevant to what you’re looking for belong to some AMS series:

- Isaacs seems to cover all and more, and has glowing reviews http://www.ams.org/bookstore?fn=20&arg1=tb-aa&ikey=GSM-100

- Rotman ditto http://www.ams.org/bookstore-getitem/item=GSM-114

- Vinberg does most things too http://www.ams.org/bookstore-getitem/item=GSM-56

And then a bit further off:

- Sepanski does groups, fields, Galois but little or no rep http://www.ams.org/bookstore?fn=20&arg1=tb-aa&ikey=AMSTEXT-11

- Aluffi is the one already mentionned http://www.ams.org/bookstore?fn=20&arg1=tb-aa&ikey=GSM-104

Elsewhere we have:

within Springer’s Universitext series:

- Lorenz does Fields & Galois http://www.springer.com/mathematics/algebra/book/978-0-387-28930-4

- Steinberg does rep theory of finite groups http://www.springer.com/mathematics/algebra/book/978-1-4614-0775-1

- Kosmann-Schwarzbach does rep theory both finite and Lie http://www.springer.com/mathematics/algebra/book/978-0-387-78865-4

within Spriger’s GTM series:

- Grillet (2007) is their latest on Algebra but doesn’t do rep theory it seems http://www.springer.com/mathematics/algebra/book/978-0-387-71567-4

- Lang was reedited in 2002

- Escoffier (2001) is Galois only http://www.springer.com/mathematics/algebra/book/978-0-387-98765-1

within Princeton U press:

no algebra textbooks.

within Cambridge U press:

- James & Liebeck do rep and stuff, but without Galois http://www.cambridge.org/fr/knowledge/isbn/item1113879/Representations%20and%20Characters%20of%20Groups/?site_locale=fr_FR

- Lu & Pahlings do rep of fnite groups with GAP http://www.cambridge.org/fr/knowledge/isbn/item2714038/Representations%20of%20Groups/?site_locale=fr_FR

- Bhattacharya & Jain & Nagpaul do groups, rings Galois but no rep http://www.cambridge.org/fr/knowledge/isbn/item1145874/Basic%20Abstract%20Algebra/?site_locale=fr_FR

I haven’t searched more, hope this helps…

I did a search on MAA Reviews (http://www.maa.org/maareviews) for books published since 2005. It found 46 books, but most are not textbooks. (Using advanced search, +algebra -linear in title, pub date after 2005, subject algebra.) Many of the ones listed above appear and have reviews that might be helpful. Not mentioned above are:

- Carstensen, Fine, and Rosenberg (de Gruyter), which claims to have applications to crypto and algebraic geometry

- Knapp has a Basic/Advanced Algebra pair (Birkhauser), which starts with linear algebra and makes some different choices.

There are reviews of those and of most of the others that have been mentioned so far: Rotman, Aluffi, Lorenz, Grillet, Isaacs, Jacobson, Rowen. Might be worth a look!

This is probably off-topic, but as James-Liebeck has been mentioned, I can’t help pointing out that I consider the approach taken in this book rather inappropriate (tensor products are defined using bases only, there is too little representations and too much characters, Young tableaux are absent, and there is too much focus on classifying groups of certain orders).

Sadly, Rotman is missing most of representation theory.

I’m a big fan of Isaacs, which I indeed first saw taking the 2nd semester grad algebra course at Michigan. It doesn’t have so much of the abstract linear algebra, and in particular doesn’t cover tensor product; but I think it reasonably covers the other topics you name. (I don’t recall offhand if it has Dedekind domains, but it does have the rest.)

Isaacs is a great expositor, and I find the book to be both precise and readable.

Dummit and Foote is also a reasonably nice book. IIRC, it covers more topics than Isaacs, although I think Isaacs is at a somewhat higher level.

I’ve found Artin’s book to be more difficult to read.

Here are some books that I like as references, together with tables of contents for some of them. No one of them fits all of what you’re looking for, and some are not even in English, but I figure maybe it’s interesting to mention them at least.

* Steven Roman, Advanced linear algebra.

http://www.springer.com/mathematics/algebra/book/978-0-387-72828-5

This is a (very excellent) linear algebra reference, but it has a few chapters on module theory which are quite well done.

* Dino Lorenzini, An invitation to arithmetic geometry.

http://www.math.uga.edu/~lorenz/book.html

This book goes over quite a lot of commutative algebra and number theory.

* Jean-Pierre Serre, Linear representations of finite groups.

* Pierre Colmez, Éléments d’analyse et d’algèbre (et de théorie des nombres).

http://www.editions.polytechnique.fr/files/pdf/TDMD_1587_9.pdf

* Daniel Perrin, Cours d’algèbre and the corrsponding book of solutions to exercises by Pascal Ortiz, Exercises d’algèbre are staples in the French system. They are undergraduate level, but full of robust and interesting examples.

I wholeheartedly recommend Jacobson’s Basic Algebra I and II. In my graduate algebra second term, we mostly followed Basic Algebra II. Most of what you want to teach is in Volume II, and in particular the representation theory does not depend too much on the chapter on modules (just a few short theorems about completely reducible ones, etc.) which can easily be incorporated into the flow of the representation theory.

As they were republished from Dover, you can get both for 40$, which is less than the other books. As a graduate student, it is the only “basic algebra” book I still refer to, as it is more comprehensive, covering things like infinite Galois theory, valuation theory, and Dedekind domains.

I think Dummit and Foote, while being a nice book, is unsuitable for the kind of graduate course you describe. It is overly wordy and doesn’t encourage categorical thinking. It also is not very useful after a first course, not covering many advanced topics. Artin’s book is less comprehensive than Dummit and Foote, seems a little to informal, and is much more suitable for an undergraduate course, although it has plenty of nice examples. Rotman looks nice and has some interesting selected topics, but is still less comprehensive than Jacobson and more expensive.

Aluffi’s book is more categorical, but it seems to gloss over lots of basic algebra including representation theory in exchange for an unusually (for a grad algebra text) comprehensive coverage of homological algebra including a bit about derived categories. In my mind the homological algebra seems too much, because if you are going to go to this depth you may as well supplant the course with Weibel’s book, which probably cannot be improved upon.

As for readability and clarity, Jacobson is first-class in my opinion, and he doesn’t add sophistication just for the sake of it, or sacrifice sophistication to placate the impatient.

It’s nice to have an excuse to rant about graduate algebra textbooks, now that I’m not teaching anymore (and never wrote any of those). Actually, I taught basic graduate algebra courses rarely (partly because they were much in demand by other faculty).

It’s not clear to me that any single textbook makes sense, in fact, because the big ones (like Dummit & Foote) cover far too much material and have their own private agendas (such as number theory). The real problem in this kind of course is to find important topics that help to convey the flavor of advanced research. It doesn’t make sense to pack into a course “everything you might someday need to know”. Way too much stuff. Even in Jacobson’s books, which I do like for reference, having been a student in some of his courses. His tendency to write in very long paragraphs can be forgiven, but his soft southern accent is absent in the books.

For example, routine coverage of Sylow theorems or finite group representations won’t be of any practical use to many of the students but may well introduce them to important ways of thinking. On the other hand, modules over PIDs illustrate well the power of generalization: finitely generated abelian groups meet essential linear algebra involving canonical forms.

At UMass, the general increase in graduate student preparation has finally made it possible to do some serious group representation theory in the course taught here. Whatever the student eventually does, in pure or applied math, that subject has the big advantage of being a meeting place for many different kinds of ideas. Not just a deducing of narrow facts from narrow axions, as in some traditional branches of algebra.

Final rant: the kind of exercises studets do (singly or in groups) counts more than the choices made in a single textbook. In graduate school teaching from the book fades away, or should.

It is a matter of no technical importance whatsoever, but I could not help but note that the date of this post — October 25, 2012 — just happens to be the 201st anniversary of the birth of Evariste Galois!

I’d say that among books mentioned, Vinberg’s is my favourite (although I can read it in original language). One advantage of it is that’s short enough (being less that 600 pages) to be read completely, unlike many other books. Dummit&Foote is too long and boring, don’t point a punchlines well, while Lang is hard to read… Knapp explains some things quite nicely, but is very uneven at some other places.

I would like to upvote Aluffi’s textbook. It is incredibly clear, especially on Galois theory (why “separable”, why “normal”, why “Galois”?). Aluffi often tells us about the difficulties he had learning the subject and really tries to explain in detail crucial examples and insights. It is very original, in this level of honesty, very modern -post-anti-bourbakist if I may. It reminded me of Tim Gowers’ posts. And the book is just handsome, like most of the AMS’s -a good opportunity to give them well-deserved money.

Definitely look at Knapp IMHO.

It seems you decided to use Dummit & Foote. May I ask why you favoured this text over the others mentioned above?

Dummit & Foote is very good book, but a few books I prefer

1. E. Artin Galois Theory: Lectures Delivered at the University of Notre Dame.

If you want to cover representation theory, you can ask your students to do semi simplicity from 2. Lam’s First Course in Non Commutative rings.

For representation theory, I don’t think there is alternative to 3. Serre’s Linear Representation of finite groups.

Even though I am fan of Serge Lang’s book, I personally don’t prefer his graduate Algebra text book. But once you cover Galois Theory from Dummit, I think the Field and Galois theory from Serge Lang is must do.

And finally, David Cox Galois Theory is another nice book.