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I have read Artin's group theory and linear algebra, Dummit and Foote's group theory and I recently come to know about Contemporary Abstract Algebra by Gallian.

I want to start reading about Ring theory and Field theory and also clear my understanding on these topics. I will be self studying it on my own but I want to get the best understanding.

I want to do a book which has hard exercises but gives a lot of thinking. All these books seem to have 50 exercises after each chapter and doing all of them doesn't seem the best use of time as many exercises use similar ideas.

Any books you would like to recommend? Or any advice on how should I read summit and footer or Gallian?

I have gone through the posts at here and here. However, none of them talks about the exercises problems perspective.

Raheel
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  • What about this https://math.stackexchange.com/questions/4927194/which-one-of-these-books-is-better-for-abstract-algebra/4927201#4927201 – Bowei Tang Jun 25 '24 at 06:53
  • "I want to do a book which has hard exercises but gives a lot of thinking" -- if you choose Artin, you may prefer to do exercises in the 1st edition as the exercises in the 2nd edition are a bit watered down; in terms of pace and exercise selection you can also mimic the course taught by McMullen if you are so inclined http://people.math.harvard.edu/~ctm/home/text/class/harvard/123/02/html/index.html [rings were done at the end of 122 right before this] – user8675309 Jun 26 '24 at 20:11

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For the Ring Theory part, I would certainly recommend Michael Artin's Algebra chapters. This book is particularly valuable because it provides a swift introduction to some concepts in Algebraic Number Theory. For example, he introduces unique factorization in the imaginary quadratic field, the ideal class group, and discusses some Diophantine equations, among other topics. Additionally, it serves as a readable introduction to the basics of ring theory.

I understand your concern regarding the number of exercises. When I was self-studying ring theory, I typically did one computation-type exercise and selected three to five proof-based exercises per chapter before moving on to the next.

For the Field/Galois Theory part, I didn't use Artin. Instead, I used Field and Galois Theory by Patrick Morandi. It is very readable, not overly dense, and includes good exercises for self studying.

P.S. I would have written only a comment, but I don't have enough reputation.

dren's theorem
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