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I am a high school student who is trying to learn abstract algebra. I know that there are many threads on this already but, I have been through them and am stuck. I have been trying to get a good grasp on abstract algebra for close to two years now.

Here are some of the sources I have been through:

  1. I started with Rotman Advanced Modern algebra but, ended up having to switch after getting to the section about Galois theory because bunches of details were left out. I tried filing them in but, I ended up confusing myself more.

  2. I went to Knapp's Basic algebra which had many topics about algebra that I wanted to learn about but, ended up having to switch after reading the section about tensor products when details ended up being left out and I was not able to fill them in.

  3. I tried Dummit and Foote which was a really good resource. I ended up getting stuck at the part on rings when I tried to fill in some details which I could not fill in.

  4. I went around a third of the way through Aluffi which was another great resource. However, as with the others, there were details I had to fill in and I ended up only confusing myself trying to fill them in.

  5. I found Sharifi's notes on Abstract algebra to be very comprehensive and detail-oriented in the beginning but just like with the others, there were details to fill in that I could not fill in.

  6. I looked through many of the sources suggested in other threads such as Saunders Maclane, Hernstein, etc. but, I couldn't find a source that was as comprehensive as Dummit Foote or Rotman.

My favorite source of these is the one by Rotman and I am looking for a resource that covers similar topics (I was using the 2nd edition of Rotman for a while. If the third edition is more detail-oriented, please let me know). Does anyone know of an abstract algebra source that is both comprehensive and fills in the details?

Chirag Verma
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  • I have nothing to suggest about abstract algebra, but I find that sometimes I do have to go and study something else in order to learn what I want to understand. Of course you have to be careful not to get stuck reading a whole book when you only want to learn part of it, but I think you're already used to that. You also already know how to look at different sources to see which ones complement each other and are useful for you. Good. Studying A to learn about B isn't a bad thing, anyway. At some point later, you're likely to find that it was useful to have learned a bit about A. – Mars Oct 07 '20 at 04:42
  • I totally agree, thank you for your insight! – Chirag Verma Oct 07 '20 at 04:47
  • Please search for your answers before posting. You have re-posted a very common question, one which has huge numbers of responses already. – rschwieb Oct 07 '20 at 13:55

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It seems to me that your main problem is that most resources from which one can learn abstract algebra tend to eventually begin leaving certain details to the reader. I have found that this is a widespread phenomenon in the majority of advanced mathematics. The more advanced, the more tends to be left out!

I remember this being quite a problem for me when I first began to study more abstract topics (algebra, real analysis), rather than more problem based topics (calculus). There is definitely a steep learning curve between these two types of material, and unfortunately if you want to continue with mathematics it is one you must eventually climb. In doing so you will develop a lot of mathematical maturity, and find this kind of ommitted material easier to handle in the future.

In the mean time, I would suggest picking a single book as a framework, and sticking to it as much as possible. Doing lots of problems from it will help you learn how to bridge gaps in the material, and sticking to a single book will help keep you focused. Of course if you get stuck on a specific topic there is nothing wrong with seeing how other books handle it, but afterwards I would go back to the main book and try to continue.

CoffeeCrow
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  • Of course, I agree; I have noticed the same thing too. I would try to fill those details in myself and if I couldn't, I'd research or try asking questions. I only decided to switch resources when I got so stuck that I couldn't proceed any further. I tried to stick with one resource for several months and cross-reference at the same time but, I just ended up getting stuck repeatedly. I totally agree though – Chirag Verma Oct 07 '20 at 05:03
  • This is all very relatable! It really does sound like you're instituting good study practices. Beside the above, all I can think to suggest is another book and another approach. Book-wise I would suggest Algebra by Artin, for me it struck a good balance between detail and depth. Approach-wise, I would echo Mars' suggestion of perhaps trying to include your breadth of knowledge by studying some other areas for a change, rather than just your depth. – CoffeeCrow Oct 07 '20 at 05:09
  • Of course. I have been doing analysis for a while but didn't talk about that because that's been going a lot better for me than algebra. Thanks for the suggestion! – Chirag Verma Oct 07 '20 at 05:24