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I recently took a graduate real analysis course and modern algebra course at the same time, and in the former course, many of the strategies listed in Tao's art
by siraben 5y ago
I recently took a graduate real analysis course and modern algebra course at the same time, and in the former course, many of the strategies listed in Tao's article came up. However, one important caveat is that several of them are specific to analysis. Since it heavily relies on classical logic and reals, indirect arguments are often used whenever possible. OTOH in algebra, more direct arguments are favored.
This is analogous to how different programming paradigms have specific ways of organizing programs and abstracting details. Likewise, in measure theory one is at liberty to say "let f : N -> Q be an enumeration of the rationals" and carry on, whereas such a statement in algebra would likely need a more explicit construction.
- gmadsen 5y agodid you use dummit and foote? I took undergrad algebra like 10 years ago and was thinking of trying to work through that book.
- mrfox321 5y agoI enjoyed D&F. Worked through up until Ring theory. One thing I would suggest is find suggested problems based on a university course. Since some consecutive problems can be similar, you don't get as good of ROI as you would spending time on a smaller and more diverse selection.
- siraben 5y agoIt was actually entirely done through the professor’s notes. I’m not sure what the reference materials were but it seemed pretty standard (group theory up to free groups and relations, ring theory up to principal ideals).