Short proof of Abel's theorem that 5th degree polynomial equations cannot be solved

This is a shortened and slightly modified version of Arnold's proof. Familiarity with complex numbers is required to understand the proof.

Recommended further reading/viewing:

[GeneBChase]

Not only does this explain the unsolvability of the quintic, but as a byproduct illuminates the solvability of quadratic, cubic, and quartic.  What a creative approach!

[KapustaCuber]

The new video on this by channel “not all wrong” introduced this amazing proof to me, but this video’s orthogonal approach really solidified it for me. I feel like I understand the “why” behind this theorem that I’d always assumed would be out of my reach without some serious group theory and Galois theory. Thank you!

[Wagon_Lord]

I really love seeing different branches of mathematics come together like this, extremely creative and beautiful.

[Moinsdeuxcat]

This is remarkable : this is an elementary justification of the analogy between fundamental groups and Galois groups, which is at the core of etale homotopy theory.

[alexyz]

That last slide alone is extremely eye-opening. I wish people explaining Arnold's proof would begin with that before working their way down into the nitty gritty detail. Great video and explanation!

[Adityarm.]

After watching it a few times, really nice & intuitive.

[richardbloemenkamp]

I vote this video as my most appreciated video after several years of Youtube watching including Numberphile, Mathologer, El Jj, Mindyourdecisions, 3Blue1Brown, blackpenredpen, Vsauce, PBS infinite series, singing banana. I think it deserves millions of views. If ever you have time to make another math video of such quality then please do.

[eyedkingdom]

Very clear video. This is the best explanation I've seen that avoids borrowing machinery from group theory -- (at the cost of calculating a bunch of commutators).

[theboombody]

I really appreciate Arnold's method even though I still can't follow it all. At least it's less abstract. Thank you for the upload!

[ellenajt]

This is really, really nice!
...But is there some elementary way to deal with running into the ramification locus of the purported formula? In that case you don't have unique lifts of loops, and so talking about taking commutators becomes messy. (I don't think 'always taking the same lift' works, because then there's still a number expressible with choosing various branches for those radicals that moves continuously and ends up somewhere different than where it started, which breaks the contradiction. You won't end up with a contradiction to the statement that values obtained by the formula with radicals are always roots, or that there is some choice of branches that will always give you a root.)
You can deal with this by observing that the bad set of paths between ordered roots is an algebraic set, being the preimage under the roots -> coefficients map of this ramification locus for the formula with radicals, and so its complement is still connected and you can still get any desired permutation by lifting some loop. However, this takes away from the elementary nature of the argument... :-( ... so I hope I'm confused and am missing something.
(We should probably should delete the discriminant locus in the same way as well, so we genuinely have covering spaces and can lift without more ambiguity than the starting ordered set of roots. I think we should also probably take an explicit 5-cycle as our permutation in A_5, so every root gets moved and one can't complain that maybe one of the fixed points of the permutation was the thing given by the formula in radicals. I guess this depends on what the precise statement is that we are trying to show.)
Anyway -- thanks for the great explanation and video!

[Crasshopperrr]

I definitely feel inspired to make an animation like this myself though … or a Shiny app.

[swampwiz]

I would like someone to do a video about this video and explain things in more detail. I still don't get what is so important about commutators.

[erickpederson]

Hang on.... This is why there are number systems from the reals(1st degree), complex(2nd degree), quaternions(3rd degree), and octonions(4th degree), but the sedenions are not a valid alternative algebra. Basically the increase in dimensionality finally breaks symmetry too much and all structure that we typically rely on breaks down. My understanding of that immediately connected the dots with quintic functions and their roots, thank you very much for this video!

[erickpederson]

Hang on.... This is why there are alternative algebra number systems for the reals(1st degree), complex(2nd degree), quaternions(3rd degree), and octonions(4th degree), but the sedenions are not a valid alternative algebra. No valid alternative algebra = no general algebraic roots. Basically the increase in dimensionality finally breaks symmetry too much and all structure that we typically rely on breaks down. My understanding of that immediately connected the dots with quintic functions and their roots, thank you very much for this video!

[Crasshopperrr]

This is a great idea … but it would be easier to follow the visual if the coefficient points were different colours. Or if you added some trails sometimes.

[Neilcourtwalker]

So this means that a 5th degree polynomial can have solutions that can't be written by a finite number of nested roots?! But could they be written as an infinite nuber of nested roots?

[casalcantara]

This is just fantastic!!! Many thanks for this proof for dummies!

[rshell]

I don't understand why returning back of the candidate expression back to start while not returning back of roots contradicts in any ways. Can someone please explain? Edit: sorry for strange comment. For anyone else wondering same question: if candidate expression indeed solution, then movement of root should correspond with exact same movement of set of results of expression. They would move exactly same as roots, though we don't know their order. But in the end, their permutation should be exactly the same as our permutation along roots. But we show they will return back to the start -- trivial permutation. Contradiction.

[bowtangey]

The same set of roots results. Why is it significant that two (or more) of the roots changed places?

[ytsas]

2:23 How in the world did moving the coefficients in loops make the solutions swap places??