Lately I've brcome really curious about Galois theory, specially about proving that there is no solution by radicals for polynomials with degree $5$ or higher. Since I know very little about group/field theory I've had to understand a lot of new notation and get my head around many new concepts.

I'm currently trying to read the six page proof Galois theory for beginners but it's proving to be harder than I thought. The author seems to expect the reader to make some underlaying assumptions which, if not made, could lead to serious confusions. I'm currently stuck with this document and I was wondering if maybe I'm just doing something that is harder than it should.

Many youtube playlists prove interesting things, although I haven't found one that proves the Abel-Ruffini theorem. All the blogs I've seen that attempt to provide a proof for people who know practically nothing of group/fiels theory fall short (the authors seem to have abandoned them halfway through). And I have heard of books that could work, although with work and college I would like to leave this as the last option.

My question then is, has someone by any chance encountered a complete proof of the Abel-Ruffini theorem that could be understood by a beginner in group/field theory?

I would truly appreciate any thoughts!

I understand that my question is not really about a math problem, but I figured that the people from this site might be the best ones to ask.

  • $\begingroup$ Did you study the splitting field / Galois group of cubic polynomials ? This is clearly a prerequisite and there are a lot of questions on MSE detailing the different cases. The next step is to study some cyclotomic and abelian extensions of number fields and see how radical extensions are contained in tower of abelian extensions. $\endgroup$
    – reuns
    Sep 21, 2017 at 15:33
  • $\begingroup$ There is more preliminary work than you might guess. You could take an entire abstract algebra course, and when you were done, you would be ready to begin Galois theory. You need some group theory. An explanation why the group $A_5$ is a "simple group." And an introduction to fields, and you are ready to start to tackle Galois theory. There are entire books on abstract algebra you can download, and lectures on the subject on You Tube. I have looked at some of the lectures by Benedict Gross on group theory that I liked, but I don't thing he gets to Galois theory. $\endgroup$
    – Doug M
    Sep 21, 2017 at 15:34
  • $\begingroup$ @reuns. I have studied the splitting fields and galois groups of cubics in youtube videos. No idea about the meaning of "cyclotomic" (I'll google it), and I have worked a little with radical extensions. $\endgroup$
    – Leo
    Sep 21, 2017 at 15:43
  • $\begingroup$ Possible duplicate of math.stackexchange.com/questions/720630/… $\endgroup$
    – lhf
    Sep 21, 2017 at 15:52

3 Answers 3


I suggest that you read Niels Hendrik Abel and Equations of the Fifth Degree by Michael Rosen for a proof of the Abel-Ruffini theorem.


Try these books:

and this expository paper:

See also these answers: 1, 2.


I would recommend Ian Stewart's book, this reads like a novel (it has a lot of historical facts) and introduces you to all the algebra you need. It comes with a set of very nice exercises.


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