What is...the Abel-Ruffini theorem?

Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.

This time.
What is...the Abel-Ruffini theorem? Or: Loops and roots

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Slides.

Just to be sure.
I shouldn’t have offended anyone! Many people would claim that Abel’s proof is very different from Arnold’s.

Material used.

More material.

Nested radical.

Galois group of X^5-X-1.

Linear, quadratic, cubic etc. equations.

Abel, Arnold and Ruffini.

History of algebra.

Mathematica.

Other online animations.

[nobodysfool]

At 10:00 and at 19:30 🤯 “as the coefficient b moves continuously in a loop (around the critical point) the roots do not move in a loop, they SWAP”

[akbarwicaksana]

Absolutely loving this channel, thanks so much!

[SRangarajan]

Can you please explain why one commutator isn’t enough to disprove nested roots? Nothing picks up a phase right? Then how does nesting help?.

[freshmanjoe]

Was watching anime a pop up ad opened and it took me here

[idilmetin]

Thank you for this fantastic explanation. I am researching the quintic formula but all of the other videos and papers I found were above my level of understanding. Thanks to your video I now understand the Theorem better. Best of the day to you because you sure made mine better. :) (P.S.: So sorry for the mistake. Still have a great day!!)

[theboombody]

This topic is very complex with a lot of ugly vocabulary. Just like this channel here, I also made a video trying to explain it in the shortest, most intuitive path that I can think of, although I’m no expert. Here’s the best idea behind it I've got:

1) A quintic formula, if it existed, would have to be able to have all 5 of its roots interchangeable, since Vieta’s equations show that the coefficients in such a formula would be symmetric. ALL FIVE roots would be interchangeable.

2) It’s possible that a quintic equation may have a root with another root nested inside of it. Perhaps more than one. This could cause huge complications with interchangeability. Each radical root would have conjugates, and swapping a root that’s nested in another root would really mess up the conjugates of the outer root if done carelessly.

3) To avoid this potential issue, the set of all roots of the polynomial would have to be arranged in such a way that no matter HOW you permute the entire set of all five of them, the subset of the roots nested inside of other roots would have to stay in its own set. Any set of roots like this fits the criteria of a normal subgroup. There may be normal subgroups that do not have this property, but if a set DOES have this property, it will be invariant under conjugation, just like a normal subgroup.

4) The nesting of radicals can be thought of as the nesting of subgroups. And every time there’s a set of roots nested inside of another set of roots, it would have to obey the property of staying inside of its own set, no matter how the full group is permuted.

5) If a group of roots is not a chain of normal subgroups ending in the trivial group, the property of roots staying in their own group is not guaranteed. S5, the group representing all possible arrangements of 5 elements (roots in this case) can possibly have subgroups that are not a chain of normal subgroups. The A5 group is one such subgroup. Since this is a possibility, the quintic is not solvable in radicals.

[devrimturker]

So what does Alternating Group A5 represent?

[RooftopDuvet]

Wonderful. Thanks for sharing this.

I've watched plenty of videos about Galois theory and I feel like no-one has mentioned the fact that you can physically make the roots undergo a permutation like this. It's so much clearer how a phenomenon like this could motivate someone to codify these permutations with symbols, and how that could then later give rise to notions of fundamental groups etc too.

Do you happen to know how familiar Galois was with Abel's work, or this visual movement of the roots? Or was his intuition purely algebraic?

(Edit: I just realised that you said that this particular proof was more from Arnold. Does that mean that this method of visually moving the roots wasn't really a tool employed in the 1820s?)