Why you can't solve quintic equations (Galois theory approach) #SoME2

An entry to #SoME2. It is a famous theorem (called Abel-Ruffini theorem) that there is no quintic formula, or quintic equations are not solvable; but very likely you are not told the exact reason why. Here is how traditionally we knew that such a formula cannot exist, using Galois theory.

Correction: At 08:09, I forgot to put ellipsis in between.

Video chapters:

00:00 Introduction
00:23 Chapter 1: The setup
04:38 Chapter 2: Galois group
11:15 Chapter 3: Cyclotomic and Kummer extensions
19:43 Chapter 4: Tower of extensions
27:25 Chapter 5: Back to solving equations
35:23 Chapter 6: The final stretch (intuition)
43:25 Chapter 7: What have we done?

Notes:

I HAVE to simplify and not give every technical detail. This is made with the intent that everyone, regardless of their background in algebra, can take away the core message of the video. This can only be done if I cut out the parts that are not necessary for this purpose. As with my previous video series on “Average distance in a unit disc”, this is made to address the question I always had when I was small - treat this as a kind of a video message to my past self.

The reason we have this mess is that we defined Galois extension using the splitting field of a (separable, i.e. no repeated roots) polynomial. The usual definition given is exactly as above - only things fixed by ALL automorphisms over Q must be in Q itself. This typical definition will of course solve the problem above, but will now create the problem of why this definition implies the larger field is made by adjoining the roots of some polynomial. These two definitions are equivalent, but I just think that it makes much more sense to define it the way I did in the video, in the context of the video; and also I think this is an easier definition to accept.

[The question is already the proof - it is a really elementary way to show the result that we want]

More resources on proofs that A_n is not solvable:
[You need to only go up to Page 5 towards the end of the proof of Theorem 2, but you definitely need group theory lingo]

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[mathemaniac]

Correction: 36:31 S_5 is the symmetric group of degree 5, not order 5.

As a result, during the peer review process, if I am your video's intended audience, then I would do my best to give constructive comments on the video. I highly encourage viewers like you to check out all the videos with the #SoME2 tag, and give all those video creators some love.

[blackpenredpen]

Your presentation is always clear and relaxing! I have to spend more time later in order to understand this video. Great work as always!

[DaveJ]

Brilliant exposition.
And, as it happens when it's about real mathematics, only a few numbers were mentioned: 1, 2, 3, 4, 5.

[Mutual_Information]

Evariste Galois was so impressive.. especially considering the fact that he invented all this before he died at 20 years old

[johnchessant]

When I was learning this I couldn't really figure out how to explain this to anyone who hadn't done group theory yet, which is a sad fate for a subject as beautiful as Galois theory. Kudos to you for explaining it so well!

[PunmasterSTP]

I definitely have to invest more time until I can thoroughly digest all of this information. But I think this video has already helped me immensely in my quest to get there. Taking a step back, it seems incredible that this is available for free on YouTube. Thank you so much for taking the time to make such top-notch material, and I hope that you keep up this amazing work for the foreseeable future!

[mathemaniac]

A bit of remark:

I HAVE to simplify and not give every detail. The intent of this video is to not dumb it down too much, but at the same time not give every technical detail so that it is still accessible. The final bit of (a) why S_5 is not solvable, and (b) why any particular polynomial has Galois group S_5, are dealt with by intuition, and I do expect people to come unsatisfied with this. However, I still leave out those details because it uses more group theory than I would like to include in the video (actually it is also because I have a bit of crisis making such a long video).

For (a) in particular, if you know group theory and the proof, I hope you agree that group theory is only slightly more civilised than "brute force" - essentially those constraints allow you to brute force everything, but group theory allows you to skip quite a bit of calculations, but it still leaves you with quite a few cases you need to deal with. In fact, I have actually flashed out the sketch of the proof on the screen. For people who don't know group theory, it will feel as though somehow magically things work out in S_5, but it does not answer the "why".


For (b), it starts with theorems in group theory (and ring theory) to get you started, but ultimately it is still a bit of fiddling things around and again magically the Galois group is S_5. So again, it would not answer the "why", and so I appealed to intuition saying that most quintics are not solvable.

As said in the video, if you want the details, go to the links in the description; but honestly the best approach would still be studying group theory in more detail.

But in any case, I do hope that you are motivated to study group theory because of this - but I have to be honest, don't study Galois theory JUST because you want to know this proof in more detail. Galois theory is difficult, and it is actually pretty ridiculous and ambitious for me to even attempt to make this video. Study Galois theory only if you are really into abstract algebra and like playing around with these abstractions.

[robokaos]

Abstract math is difficult for me. I appreciate high quality videos such as yours to help mitigate that struggle :)

[PowerhouseCell]

Beautifully done! Cannot even begin to imagine how much time and work went into this. Manim's difficult to use but you did it incredibly well!

[charleschidsey]

I’m almost speechless. Thank you so very much for this brief (by necessity) introduction to Galois theory. You have earned my subscription. Best of luck with your channel going forward.

[Adityarm.]

This was amazing, thanks a ton! To summarize:
- The process of solving equations via +-*/ & radicals is equivalent to starting with a base field of accessible elements, & then including new layers of numbers [which are roots of the currently accessible field elements].
- This [cyclotomic+Kummer] extension tower has a very specific property, that the symmetries of the newly included numbers over the previous layer always contain the previous symmetries as commutative-normal-subgroups.
- Alternating group A5 has no non-trivial normal-subgroups, it's the smallest non-commutative simple group. We run into this when looking at S5 symmetry of some quintic equations.
- The previous points imply that extending layers of radical expressions of field elements can never reach quintic structures.

Please correct me if I'm missing anything here.
This sounds very similar to a high level sketch for proving which numbers are constructible [only the ones which we can reach through tower of field extensions of degree 1 or 2]. There is an arxiv which also has a beautiful representation of A5, Galois Theory : A First Course - which, as the author explains, coincidentally looks like the simplex known form of Carbon :)

[Ben-x]

You made this about as clear as possible without a full course. I have definitely gained insight into this difficult area.

[AllemandInstable]

one of the ( if not the ) best channels of maths out there
i love that you deal with topics which are at a higher level than most of math content on youtube

[LucasDimoveo]

I hope this channel gets more views. The editing and audio quality are fantastic

[cmbryant]

Abstract algebra has always been a weak point for me. However, since I am interested in algebraic topology, I figured I should get comfortable with computing the homology and cohomology groups. My ultimate goal is to study the application of Lie groups to differential equations. I was inspired by the Frobenius Thm, and it just sort of clicked.

Picking up abstract algebra again, I find I am engaging with the material in a new way. Exact sequences led me to study normal subgroups and quotient groups. But now, I can also see the significance of permutation groups. Your video gave me a quantum leap into the Galois theory endpoint. Galois groups motivate the prerequisite material very nicely.

[ddSavant]

I love seeing stuff like this on YouTube. A lot of people think computers are the be all end all of mathematical processing - and that’s true, to a point. Computers are phenomenal at simple operations, and sorting. Computers are not any kind of good at abstract mathematics. They’re slowly getting better, but they don’t have intuition, and they aren’t able to substitute or generalize well.

People who can do complex and abstract math are never in huge numbers, but are badly needed for scientific advancement.

Keep being awesome.

Cheers!

[orpheus]

Thank you for the time and effort put on the video. Channels like yours make the world a better and more interesting place to live on.
Fascinating topic and a beatiful example of the abstration capabilities of human reason.

[milos_radovanovic]

Good addition to 'Aleph 0's and 'not all wrong's videos. This topic definitely needs more exposure and expositions!

[MatildaHinanawi]

This is SO hard but I've been so curious about it for ages, props on the video!

[redflame]

For a man who was alive for such a short time, Galois truly did manage to live forever 🙏🙏