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I took an introductory abstract algebra course at my university and was fascinated by the content. I would love to learn more and go into greater depth with groups, rings, and fields, but unfortunately there aren't any other algebra classes where I attend. If it all helps, the class I took covered basic ring theory through the lens of number theory (we also covered some polynomials rings as well) and (extremely) basic group theory regarding subgroups, cyclic groups, and permutation groups. With summer coming up, I figured I could take the time and self-study as much as I could. In the end, I would love to understand the Abel-Ruffini theorem and the nitty-gritty details as to why polynomials of degree 5 or greater are unsolvable by radicals.

To give an explicit question: what are some resources that I could potentially use for self-teaching abstract algebra and what should I focus on so that I can work up to understanding (i.e. being able to prove myself) the insolvabilty of the quintic?

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I will sketch the proof of the Abel-Ruffini theorem by describing how one would show that the roots of $ X^5 - X - 1 = 0$ are not expressible using radicals.

  • Show that a radical extension of $ \mathbb{Q} $ has a solvable Galois group. This is because of the fundamental theorem of Galois theory, which tells us that there is a one-to-one, inclusion reversing correspondence between Galois subextensions of an extension and the normal subgroups of its Galois group.
  • Show that the symmetric group $ S_5 $ is not solvable. Its only nontrivial normal subgroup is $ A_5 $ which is simple, so its only composition series is $ \{e\} < A_5 < S_5 $. However, $ A_5 $ is not abelian, so this is not a valid series and $ S_5 $ cannot be solvable.
  • Show that if a subgroup of $ S_5 $ contains a 5-cycle and a 2-cycle, it is actually the entire group. This can be done with a basic proof, see Why is $S_5$ generated by any combination of a transposition and a 5-cycle? for an example.
  • Show that $ X^5 - X - 1 $ is irreducible in $ \mathbb{Q}[X] $. This can be done by noting that it has no roots in $ \mathbb{F}_5 $ and it cannot split into quadratic factors in $ \mathbb{F}_5 [X] $ either, because then it would have a root $w $ in $ \mathbb{F}_{25} $ and we would have $$w^{25} - w = (w+1)^5 - (w^5 - 1) = (w+1)^5 - w^5 + 1 = 1 + 1 = 2 \neq 0 $$ which is impossible.
  • Show that the Galois group of $ X^5 - X - 1 $ over $ \mathbb{Q} $ is $ S_5 $. To do this, note that the Galois group must be a subgroup of $ S_5 $ as it will permute the roots of the polynomial. Now, observe that the polynomial has one real root, therefore complex conjugation is an automorphism of the splitting field. Furthermore, observe that the quotient $ K = \mathbb{Q}[X]/(X^5 - X - 1) $ can be embedded into the splitting field, say $ L $, so that we have

$$ [L:\mathbb{Q}] = [L:K][K:\mathbb{Q}] = 5[L:K] $$

  • The order of the Galois group equals the degree of the field extension, therefore the order of the Galois group is divisible by 5, a prime. By Cauchy's theorem, it then has an element of order 5, or a 5-cycle. Since complex conjugation is a 2-cycle, by the above results this proves that the Galois group is $ S_5 $.

This shows that the splitting field of $ X^5 - X - 1$ (the field which contains the roots of the polynomial) is not a radical extension of $ \mathbb{Q} $, so the roots are not expressible using radicals. Note that this result is actually stronger than the Abel-Ruffini theorem, which is proven by introducing successive transcendental elements into $ \mathbb{Q} $ and considering the general polynomial which has these as its roots, and provides no specific examples of a polynomial whose roots cannot be so expressed. On the other hand, if a general formula existed, then it would clearly solve $ X^5 - X - 1 = 0 $, which is impossible.

To fully understand this proof, you need to be able to follow all of the steps I have outlined, which requires you to know, or at least be familiar with:

  • Galois theory
  • Group theory
  • Properties of symmetric groups
  • Field theory

For any of this, it is important that you have a solid foundation in the basics of abstract algebra. As a specific resource, you might try Fraleigh's A First Course in Abstract Algebra.

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  • $\begingroup$ Thanks for the in-depth response! I was actually looking into Fraleigh's book already, and trying to create a list of problems to do based on what other courses I could find online had done. $\endgroup$ – Ethan Hunt May 11 '16 at 13:10

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