Why There's 'No' Quintic Formula (proof without Galois theory)

Feel free to skip to 10:28 to see how to develop Vladimir Arnold's amazingly beautiful argument for the non-existence of a general algebraic formula for solving quintic equations! This result, known as the Abel-Ruffini theorem, is usually proved by Galois theory, which is hard and not very intuitive. But this approach uses little more than some basic properties of complex numbers. (PS: I forgot to mention Abel's original approach, which is a bit grim, and gives very little intuition at all!)

00:00 Introduction
01:58 Complex Number Refresher
04:11 Fundamental Theorem of Algebra (Proof)
10:28 The Symmetry of Solutions to Polynomials
22:47 Why Roots Aren't Enough
28:29 Why Nested Roots Aren't Enough
37:01 Onto The Quintic
41:03 Conclusion

[alecbg]

Great video! Beautiful idea explained very well. I was going to ask if there were functions that worked for the quintic, but Google has already told me that elliptic functions do the trick

[MayawireCH]

As someone who’s been watching 3B1B, Numberphile, Michael Penn, and the likes for years, I can confidently say that this is one of the best math videos I have ever watched. The argument is elegant and presented in a way that makes it very accessible, the animations are fantastic, and you can clearly see that you have a passion for this subject. Please keep making these videos if you can, it’s a gift to all the people out there without access to a course in abstract algebra that we can still try to learn and understand this.

[adsoyad]

16:34 "Algebraic expressions are limited in how multi-valued they can be." This sentence was so eye-opening for me

[jacoblojewski]

It wasn't until watching this that I realized why some books/texts etc. prefer to state the Fundamental Theorem of Algebra as "Every polynomial (with complex coefficients) with degree at least 1 has at least one root (in the complex numbers)". Makes it easier to not have to discuss the counting of repeated roots.

[nsfeliz]

i have found a formula for for quintics but the margins of the comments section wont fit.

[druid]

Outstanding video. An unusual approach to the subject, appealing to the imagination. Solid presentation. I had the pleasure to evaluate your work in the peer-review stage of SoME1. And it turned out to be the best of more than 20 shown to me by the system. We are, in a sense, rivals in this competition. But competing in such a company is an honour for me.

[PaulMurrayCanberra]

My first thought was, how can there be a quadratic equation? If you just use the positive root, then this gives you a smooth, single-valued function for one of the solutions. But that's exactly what this is all about: it doesn't work for complex numbers. There isn't a way of choosing "the positive root" that's smooth, because complex numbers form a plane. I heard a while ago that points on a plane "couldn't be ordered" - didn't really know what that meant. This "flipping the roots" animation … I have only just now got it. An ordering (in that sense) means that the greater than/less than relation of two points stays the same when you smoothly move them. Only works on a line.

Nice when things gell.

[janszwyngel]

I think the best part of this proof, and the main advantage over the Galois theory approach, is the fact that it doesn't use field theory. The only assumptions about a hypothetical quintic formula are, as far as I can tell, that it's made from continuous functions of the coeffitients which can be single-valued or multi-valued, but walking along a path that doesn't loop around the origin can't change the value (also by shifting the domain this can be extended to looping around any point, but not many points at once), which is much more general than the four operations together with natural degree radicals that you get from the Galois theory. It's beautiful that such a great result can be motivated by arguments from continuity and the limitation on multi-valuedness of the functions used only. The presence of the seemingly arbitrary boundary of fifth degree only makes it better. Great argument, clear presentation, and one of the weirdest, unexpected results.

[DanielKRui]

When the commutators started showing up and the commutators of commutators my brain was forming thoughts that my mouth wasn’t ready to put into words, and then when the terminology “derived subgroup” was revealed EVERYTHING clicked. Absolutely brilliant, making this very abstract math intuitive. It’s like the notion of calculable constants in physics I just read about today on Stack Exchange; the more fundamental or deep a theory, the more "standard" constants are explained from more fundamental and deeper constants. Derived series were always one of those “unexplainable constants”, but now with this deeper and more fundamental understanding, it is “merely” a corollary to a clever trick doing and undoing paths.

And my god I just got to the proof of the non-solvability of all S_n or A_n for n>=5. Absolutely stellar. That trick of expressing sigma and tau themselves as a commutator of 3-cycles felt like a magician dropping the bottom out of a box forming a well of infinite regress...I can't believe such a "simple" observation shows that all derived subgroups of S_n or A_n for n>=5 must contain a 3-cycles. I wonder if there is a nice proof of the simplicity of the A_n for n>=5 given these facts (since all proofs I know of that are still have somewhat of a "coincidental" quality to them).

In 45 minutes, you have drawn back the curtain on a semester (or perhaps even semesters!) of graduate level abstract algebra, and indeed I have waited for a video explaining this theorem/argument for a very long time (I did not understand any existing videos/papers trying to explain it); and for that, this will rank in my mind as one of the greatest math videos/expositions of all time!

[mCoding]

Fantastic explanation demonstrating a deep understanding of the material. I would argue that you did use Galois theory in this presentation, but done so intuitively as to present the concepts while avoiding the usual symbolic definitions.

[peterhall]

I'm a 70 year old functional analyst and I'm not knee deep in group theory but this is a really insightful exposition. I was not aware that Arnold cooked this approach up but in a sense it does not surprise me. His work on the geometry of dynamical systems (KAM theory etc) is full of really fundamental insights. Well done.

[johnchessant]

This is absolutely brilliant!! So glad this got featured in 3b1b's recap video.
This really gave me a sense for why commutators are important. If I'm understanding correctly, for any rational function r in the quintic's coefficients, we have a homomorphism S_5 -> Z which describes the phase difference of r as we swap the roots; the commutators are exactly the elements that map to 0 since Z is abelian!

[Adam__]

What a hidden gem this video is — Great pacing, explanations, and graphics. Thank you for making it

[helman-tov]

I watched this video earlier this week, but I came back to say it *blew my mind*.

I have never even heard about the existence of this proof before, and now I can't believe it isn't more famous.

This video is great! It's so clear and interesting.

Thank you so much for making this video. I really hope you continue to create this kind of videos once in a while.

[alphansahin]

That's a great one! For me, the critical aspect was around 27:20 in the video (i.e., Exercise 1 in Leo Goldmakher's paper, "the image of f(p)^a for any rational a as p traverses the commutator loop [γ1, γ2] is a loop in C.") In a way, a commutator allows us to discuss the existence of a single-valued function that includes a root operation.

Some MATLAB code on commutator for anyone interested:

clear all
close all
clc

if(1)
L = 4; % quartic
L = 5; % quintic
actions = perms(1:L);

for nestLevel = 1:4
Nperm = size(actions, 1);

com = zeros(Nperm^2, L);
for j = 1:Nperm
for k = 1:Nperm
L1 = actions(j, :);
L2 = actions(k, :);
com(Nperm*(j-1)+(k-1)+1, :) = fcnComm(L1, L2);
end
end
actions = unique(com, 'rows');

size(actions, 1)
end
end

function Lcomm = fcnComm(L1, L2)
[~, L1inv] = sort(L1, 'ascend');
[~, L2inv] = sort(L2, 'ascend');
Lcomm = L2inv(L1inv(L2(L1)));
end

[iwersonsch]

It was at 35:25 that I realized why that simple group in Galois theory is so important. It sounded completely unmotivated to me back then, thank you!!!

[christopherjackson]

In my studies, I had come to appreciate that the nonexistence of a quintic formula was somehow a corollary of the fact that A5 was simple.
Now that you have connected the derived series to nested radicals for me, I am in bliss and will never be the same again.
Thank you, so much!
😍🤩

[jamesthomas]

Excellent video. Great pace and clarity.
More than 35 years since I studied Galois theory and all I remember is it being hard to understand.
This proof and your explanation shows the fundamental result without all the baggage. What a beautiful proof.

[NinjaOfLU]

I can't express quite how pleasantly surprised I was by this. This is explained beautifully, paced well, and, whilst I tried to pick holes in it, I couldn't manage to: when I thought I'd found one, I paused and found that, thinking about what you'd said a second time, it had already closed itself.

Could this be done in a shorter video? Totally! And pretty easily: I bet. But I _don't_ think it could be done in a much shorter video without losing something of what makes this video so great, which is that all of the steps are motivated and intuitive, rather than simply stated. Could it be cut down? Yes. Should it be? _Definitely not._

Understanding the unsolvability of the quintic has been on my bucket list for _years_, and I thank you for being the one to finally help to get me there!

[prettyigirl]

Incredible video — despite already having learnt the Galois theoretic proof, I feel like I'm only now understanding how it works.