Algebra books with challenging exercises I am taking my algebra quals this winter, and don't feel confident with my algebra abilities based on the class I'm taking. I feel comfortable with the content that's going to be in the quals (Galois correspondence, fields of fractions, PIDs, classifying groups) but don't have much practice with applying them to harder questions/exercises. Does anyone have books that focus on basic concepts, but apply them to more challenging exercises? Something like the algebra equivalent to baby Rudin in Analysis.
Thanks!
 A: I'm not sure I can gauge whether Dummit & Foote's (Abstract Algebra) exercises are challenging (it's been a few years) but I remember enjoying them. Also my school has a qual exam archive you could work through. What I did to prep for my qual was work 60ish problems from past quals and similar quals, and it did me fine. In our program these come right after the first year so they're not super intense exams. :) Link is here: Algebra Qual Archive
A: It seems like most of the books, and I haven't read that many, but whether Hungerford, Lang (basically a classic), Herstein, I'm actually partial to Fraliegh, which I learned from under Stallings, will have hard exercises.  Maybe baby Herstein, if I can call it that, because he has a harder book, Topics in Algebra, or some such, would have problems that are a little too easy.
I think Milne has a book, though it may just be on group theory.  There are, probably, alot of more specialized books.  If you are into combinatorial group theory, I heard Magnus is good.  Finite groups, Burnside.  Representations of finite groups, someone named Bob Steinberg.  Different than Robert Steinberg, the excellent UCLA professor, who discovered the Chevalley groups.  Speaking of whom, his notes, in the back of Lie Groups, Lie Algebras, and their Representations, by V.S. Varadarajan, are quite good, as is the book itself.
One title I like is Rings and their Modules.  I forget the author, but I would like to check it out.  Modules are an important topic in Algebra.  Among other things, the structure theorem for finitely generated abelian groups is a special case of the structure theorem for finitely generated modules.
A: I don't see this book recommended often, and I honest don't know why.  But I really think Grillet's Abstract Algebra is an excellent place to learn graduate-level algebra and deserves to replace Dummit and Foote as the standard "it's not Lang" graduate algebra book.  To me, it sits somewhere between Hungerford and Lang in terms of sophistication, but it's highly readable.
It's hard to avoid Lang, and you should definitely work through some of his exercises (some of them are excellent, and some of them have.. typos).  Ken Ribet has a website with a lot of good material from courses he's taught from Lang, including some excellent problem sets.  But it's such a tome that I always felt like I learned the material better from other sources and later turned to Lang to deepen my understanding.
Grillet also has some pretty nice exercises at the end of each chapter.  Dummit and Foote has great coverage and some great exercises, but so many of them are just too mechanical, and I always walked away from it wondering if I really understood the material at the right level.  I really wish they would do some major updates to the text and cut out much of the fat.  Then again, when I was struggling to understand something from Grillet, I often found Dummit and Foote was nice for holding my hand a bit through it.  For whatever reason, I've always preferred textbooks that have a handful of well-chosen exercises at the end of each chapter, which you're expected to solve, to those with 50+ exercises which are a mix of good and bad.
That said, if Dummit & Foote made an updated edition with a lot of the chattiness cut out, better typesetting, and a nicer set of exercises, it would be fantastic.  Parts III and IV of D&F are particularly good.
