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I am going to be taking a year off from my studies and would like to self study abstract algebra as it is right now the biggest gap in my math background.

I have a copy of Dummit and Foote from which I would like to study, however I realize that it contains quite a large amount of material! I would thus like to put together a list of essential topics to cover so that at the end I would have covered a similar content to a third year undergraduate course for mathematicians. One thing I would like to do if possible is get an introduction to Galois theory, it is quite mysterious to me and I would love to get acquainted to the subject.

I am (quite unfortunately) in electrical engineering, although I am directing myself to do a masters in math or perhaps control theory on the mathematical side of things. As such I have taken as many math course as I could and have done some self studying so that I think I now have a reasonable degree of mathematical maturity (real analysis, topology, differential geometry, linear algebra of course, probability and stats, discrete math, etc). Unfortunately I can't take as many pure math courses an as a math undergrad which is why I want to self-study abstract algebra.

I know this is an ambitious project but I am quite motivated so any tips are very appreciated!

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    $\begingroup$ What is your goal? Just to get a general overview about algebra as taught in an undergraduate course? Maybe you should look at a few syllabi from different universities then. $\endgroup$ – martin.koeberl Mar 19 '17 at 16:06
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    $\begingroup$ What do you want to do with your degree? In the US it's unusual to get a master's in pure math. Basically people who try to get a PhD and fail are the ones who have master's degrees. However master's degrees in applied math are often terminal degrees. $\endgroup$ – Matt Samuel Mar 19 '17 at 16:07
  • $\begingroup$ @martin.koeberl I actually looked at the syllabus of the course from my university and thus have an idea of what to cover but since I am self studying I am a bit afraid that some topics I skipped would have been briefly covered in class, I guess my question is more specifically about self studying $\endgroup$ – Louis Mar 19 '17 at 16:11
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    $\begingroup$ Maybe a huge book like that is the wrong way to go. I learned abstract algebra by self study and I used Topics in Algebra by Herstein. It doesn't have very broad coverage, but the problems are excellent. I was preparing for an honors senior level undergraduate course. After studying the book the course was way beneath my level, so next semester I took a graduate course which was also not challenging. I could just be extraordinarily gifted at algebra, but I think it was the book that did it. $\endgroup$ – Matt Samuel Mar 19 '17 at 16:17
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    $\begingroup$ Well, if it's not in the syllabus it can't be too important. Right now, your question seems a little broad. However, you might want to look at this: extension.harvard.edu/open-learning-initiative/abstract-algebra. At least, you can check the videos to see what topics are covered in class. Also: look at the related questions on the side, there are other recommendations. $\endgroup$ – martin.koeberl Mar 19 '17 at 16:19
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I actually did this, and so I have some experience with Dummitt and Foote that I'd like to share. First, if you're not committed to D&F, but just to getting some good basis in modern algebra, I might be inclined to recommend a different book. Let me just tell you some pros and cons about this bible of algebra.

Pros:

  • Tons of examples worked out. Every chapter has some general theory, followed by usually about a half dozen explicit examples. It's really good to do these examples by yourself, and then read how the book does them, or read them in the book and then try modified examples for yourself and see if you can follow the same ideas. Another useful thing to do is to try to work some examples as you read through theorems. None of this is unique to D&F but rather just good advice for reading any math book, but it goes double for D&F for reasons I'll explain below.

  • A billion and two exercises. This is a must for algebra, for the same reason. Practice practice practice. The problems range from routine to difficult (but none are so hard I could never work them out, if you like to be $really$ challenged you may not like this about the book). The first dozen chapters are so have solutions online (google 'project crazy project'). This doesn't include the stuff on Galois theory you're potentially interested in, but it does include everything up to the stuff on modules over PID, if I remember correctly.

  • It covers nearly everything one could possibly want to know. It really is an encyclopedia of algebra, at a level that's pretty accessible - it does not have the level of formalism that other books, like the one by Lang have. No knowledge of categories is required to learn from the text, and when they are introduced in the discussion of tensor products, all the relevant notions are included.

Cons:

  • IT'S LITERALLY HUGE. Trying to finish even a substantial portion of it (let alone all) is a near impossible task. I studied from it for a year when I was an undergrad (probably less advanced than you are now though) and I only got through like 9 chapters.

  • Many of the later chapters contain things that are not part of a typical first course. The two semester sequence of algebra at my university covers only up to the stuff on Galois theory and fields, which is like 13 or 14 chapters in, and skips around quite a bit (and over some topics completely), yet the book is like 20 some odd chapters long.

  • Dummit and Foote's style is a little deceptive. They do not do a very good job demarcating which theorems are fundamental for you to understand and which are just 'results' that come up you may want to know. It is my opinion for example, that nilpotent groups, semidirect products and other content from the later chapters on group theory are not particularly important to understanding the basic ideas of group theory, the isomorphism theorems, group actions and Sylow theory, etc.

I really like Dummit and Foote's book. But it's not the be-all and end-all book that some faculty make it out to be. There are other very good books that contain other useful perspectives and are at a variety of levels. So my advice is to use D&F as a list of topics in algebra, and a great source of exercises and examples, while hunting out other references in the areas you like. Modern algebra is an ungodly large field - it's literally one of the three main branches of math - that there is no hope to learn about everything, at least at this stage. So use it as a starting point, and as a springboard into the areas of algebra you like. Ideally you could follow a syllabus from some other place, at least until you feel like you have a sense of direction, and then just peruse many references online using particular chapters or sections of D&F as a place to get started from. A given chapter of D&F is readable (with many exercises done) in anywhere from a week to two weeks, depending on your time and maturity. This is a great way to get some exposure to some ideas, and to help you think about branching out in other directions.

Here's a short list of other texts that I like for learning about algebra that may serve as useful references.

1) Chapter 0. This book is my favorite introduction to algebra. It includes many interesting topics not present in other books, and uses category theory throughout (it has a first chapter to introduce you to it) and is by far the friendliest algebra book I've ever used. As a result it's a little slow, but you can always breeze through it just for it's perspective while working from D&F. The perspective of universal properties is really essential to how algebraists do algebra, and so is immensely useful to carry with you as you learn. It's also the only book I've ever seen which is readable by an undergraduate which actually discusses any homological algebra at all - the others in this list (and D&F, if I remember right) do discuss it, but the discussion is certainly more difficult to read. There's even a short discussion at the end about derived categories and spectral sequences, which are certainly advanced topics.

2) Grillet's "Abstract Algebra." Similar to Dummit and Foote, but a little less massive. If you worked through something like this, you'd be able to get to Galois theory much quicker. On the other hand, it doesn't have the wealth of examples and problems, but the exposition is clean and clear. The later chapters go in a different direction from D&F, moving towards more homological things and ending with a chapter on universal algebra. These chapters, in addition to the later chapters from D&F on representation theory could be skimmed to see what grabs your attention.

There are a wealth of other books that are well-regarded, but I can't give much feedback on them. Rotman, Jacobson, Lang, etc. Any or all of these should be used in parallel with D&F's book in order to get the most out of all of them.

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    $\begingroup$ Great answer thank you! It is nice to hear of someone else doing self study. I will definitely try to check out the other two books you suggested. I have also been suggested Lang but as you mentioned, I think at my level it is better used in parallel with a more accessible text. $\endgroup$ – Louis Mar 19 '17 at 16:50
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    $\begingroup$ Thank you for mentionning "project Crazy project " So now I am wondering if it is possible to find to solutions to the representation theory chapters of Dummit and Foote elsewhere.. $\endgroup$ – Psylex Nov 12 '18 at 21:00

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