Textbook Questions to Do while Self-learning

Question

I am working through Dummit and Foote's Abstract Algebra this summer in preparation for a class next year. However, this is my first time really trying to learn a subject through only a text. It seems to me that doing every single problem, while probably helpful to understanding, is merely too time-consuming to make the progress I need to make. Do you all still recommend doing so? If not, how do I select exercises to do? I have already learned that skipping all exercises ensures that one learns nothing. =P

If it helps, an example, an example is the fact that there are ~40 exercises following only the axiomatic definition of a group. Should I work through all of them, or only a subset and continue on my way?

Thanks!

Answer 1

I just took a course this year that used Dummit and Foote, and a lot of the exercises in the book are pretty good and useful. I would say look over the first couple (maybe 5-10) exercises depending on the section and write some solutions for those. Then look through some of the later exercises in the section and pick out a few that you find interesting and work on those. If you find one you that confuses you, look at that one too, as their may be a gap in your knowledge.

I would say yes, doing all of them is not necessary, but there is a lot of good material in that book, and it will only help you more in the future. As someone else points out, if you are getting bored by them, then I'm sure you can move on. Many of the problems in Dummit and Foote are presented very nicely with plenty of hints to help. In addition, many definitions are introduced in the exercises, so make sure you look at those.

Answer 2

Allow me to point out the obvious fact that knowing how to solve a problem and writing out the answer are quite different. I would suggest reading the exercises/problems and seeing if you have an idea about how to do them. If you are pretty sure you know how, then just do a couple to check that your intuition is accurate. If you aren't sure how the problems would be done, then ask yourself how you might broach them initially, and try to write out a (reasonably) full solution.

Totally separately (and depending on what level you are trying to get to): I have a certain fondness for the book Abstract Algebra by Dan Saracino. Among introductory texts to Modern Algebra (which is the level I assume you are at, since you are starting with the group axioms) I find it to be quite accessible and very well-organized. The problems range nicely from straightforward/doable to tricky/puzzle-like.