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I am a graduate student.I have a firm grasp on group and ring theory.Now I am learning Galois theory.But the problem with our instructor is that he is very clumsy and I am not finding motivation behind the theorems I am doing,like what the theorem really means,although I know the motivation behind this topic which is solvability of a polynomial by radicals.I started self studying as I was not satisfied with the teaching of our instructor.But the books also do not provide much insight and simply state and prove the theorems.Although I understand what the statement wants to say but I do not realize what the theorem really wants to say.In such situation I am badly stuck and cannot find a way out.I have to somehow learn this theory properly.I also referred to some lecture videos but none of them serve my purpose.I do not have clear idea about normal extensions,separable extensions and most importantly the Galois extensions.These things seem to me coming out of the blue.I do not even understand the meaning of Galois correspondence theorem.So,I need some reference where I can find both proofs and insight behind those proofs.I want to make a complete note so that I can refer to it while studying.Can someone help me?

Kishalay Sarkar
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  • It's not very clear what kind of motivation you seek. Intuitively, the main idea of Galois theory is very simple: we try to solve difficult problems about fields (or about polynomials) using easier problems about groups. – Mark May 27 '22 at 13:34
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    This site has many recommendations for abstract algebra texts and Galois theory, e.g., here. I recommend that you do examples for each definition and each proposition. This gives you a concrete understanding. As for abstract algebra prerequisites, it seems that all your previous questions were more on analysis and not at all about abstract algebra. So you should review these notions, too. – Dietrich Burde May 27 '22 at 13:34
  • Why is this closed? In what is it different from other reference request questions which makes it opinion based? Is it only the word "good" in the title? – nadapez Feb 16 '25 at 18:53

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The book Field Theory and Its Classical Problems by Charles Robert Hadlock emphasizes the classical motivation. https://bookstore.ams.org/car-19

https://www.jstor.org/stable/10.4169/j.ctt5hh8wp#:~:text=Field%20Theory%20and%20its%20Classical%20Problems%20lets%20Galois%20theory%20unfold,polynomial%20equations%20by%20radicals%2C%20and

Yuval Peres
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  • I recommend the youtube playlist on galois theory by Richard Borcherds. It is graduate level and focus on overall presentation of the theory and examples. Very intuitive approach and clear explanations. There are a lot of other playlists by the same author on other algebra topics. https://www.youtube.com/playlist?list=PL8yHsr3EFj53Zxu3iRGMYL_89GDMvdkgt – nadapez Feb 16 '25 at 18:59