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.