Goodness, this is the big one.

I have a long and complicated relationship with Dummit & Foote. This is a text that one can get absolutely lost in, and i absolutely have. For example, i think that at one time i had solved (nearly?) all of the problems in Part I. It's full of excellent examples, it's full of wonderful exercises, and… honestly, one could probably spend the rest of one's life reading it if one wished.

That's both good and bad.

On the one hand, it's wonderful about taking its time, about being complete, thorough, approachable to students at just about any level of post-proofwriting-course experience (or maybe with an elementary number theory course under their belts). It's a text that really tries to bring everything it can to the student, and be a comprehensive guide. And… the student is well-rewarded for their efforts. This text has a lot to teach, and spending quality time with it will yield wonderful rewards.

On the other hand, because it's such a vast forest, one can find oneself lost. At the very least, it can take a /very long time/ to get to some topics, or through others. I also don't know that i find all of its explanations and approaches to be the cleanest or the most crystal-clear. For example, i've long found that the presentation of tensor products is wanting in some way that i have trouble specifying; perhaps it's that it is so general that a student can readily miss the point of the exercise, the goal of the tool. I'm not sure; i've not looked at it in some time. Mostly… i've read better presentations elsewhere.

Still… i think one would be hard-pressed to find a better comprehensive algebra text. I think Lang is probably the other major reference (and/or textbook), and… i've never found Lang's writing especially compelling—he was infamous for completing textbooks in a few months, and i think it shows in his presentation; i often find that he says too little in his texts, or tries to be too slick in his presentation in order to make topics appear simpler than they are. In so doing, he will make certain examples appear wonderfully natural and straightforward, while leaving a reader totally adrift on related problems.

I think that this is a text every student with any serious interest in mathematics should own. Even with its drawbacks, well… my copy's spine has been duct-taped back together because it has seen so much happy use, so i think its kind, gentle, and thorough presentation is well worth taking a good look at.