Galois Theory Explained Simply

[Note: as it has been correctly pointed out by MasterHigure, the dials at 8:10 should have 4 and 6 edges (as opposed to 5 and 7, respectively). That is because of each of the factor polynomials being reducible and having 1 as a common root.]

To learn more about mathematicians that made a substantial contribution to Galois Theory:

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[hannesk]

wow I'm glad the youtube algorithm showed me this hidden gem. I like your presentation and style.
Looking forward to seeing your next video!

[Yakushii]

There are some errors here.

At 7:20 you say that the Galois group of x^7 - 2 over the rationals is cyclic, but it's not. It's dihedral with an order of 14.

At 8:10 you make a similar error for (x^7 - 1)(x^5 - 1). The orders are 6 and 4.

At 12:39 you mention that the Galois group of x^5 - 2x + 1 is S5. But it has a root of 1 and is reducible to a linear and a quartic, for which there exists a formula. It's therefore S4.

[konstantinkh]

I don't know how much mileage I'd get out of this if I didn't have an undergrad course in modern algebra, but from perspective of someone who knows what groups and fields are, but never encountered Galois groups before and their relation to polynomials, this is fantastic. I just wish it'd push just a little bit further into why S2, S3, S4 can always be decomposed into a product of cyclic groups, even if just as visualization of some special cases.

[Bobbias]

Oh damn, this is really good. The combination lock idea is brilliant. The way a radical creates a cyclic group is also why the scale we use in music works the way it does. Since we use twelve tone equal temperment, each note is 2^1/12 apart. Once you stack 12 together you double the frequency. The 12 possible pitches form a cyclic group symmetry.

[amankarunakaran]

Overall I'm always happy to see more math content creators on youtube, and I'm excited to see future videos from you. Most of the Galois groups are actually calculated incorrectly here, and those kinds of details should really be corrected/verified before creating a video like this. Ignoring that, here's a few notes and nits, though (reading this over I'm worried that I'm coming off too harshly but I promise I did enjoy the video and would like to see more :) )
One big question I'm left with after seeing this is - who is your intended audience? Curious high schoolers? Undergraduate math majors? Undergraduate non-math STEM majors? Any curious undergraduates? Graduate math students/Post graduates in mathematics?
My guess is that it is meant for curious high schoolers or undergraduate non-math STEM majors (particularly because of the commuting functions analogy and notation), perhaps with some math majors as well. I think this question needs to be addressed quite carefully, because it will address the question of how much rigor your videos require, which I think was probably the primary weak point of the video (and is imo the point that most math videos on youtube struggle with, even for big channels like numberphile).
I'll try pointing out stuff as I see it through the video:
0:00-0:50 I think this is a solid intro, motivations for the subject are definitely clear. I think it's a *little* disingenuous to say that we'll "answer the question today using Galois theory", since it's really more of taking a peek at the theory that needs to be developed in order to answer the question, but all good intros are probably a little disingenuous in a clickbait-y way, so I think this is fine.
0:51-2:34 I think this is pretty good, and a fine introduction into the idea of field extensions. However, I think it could be a *little* clearer about the finite operations thing. 1+sqrt(2) is an example, but it's also probably worth showing something like It's also somewhat nonobvious that this ends up being equivalent to the set x+y*sqrt(2) for rationals x, y (division being the nonobvious part). If the intention is to just briefly wave Galois theory in front of the audience, then omitting such details is probably fine, but it's worth at least pointing to the parts that you are handwaving over to acknowledge that they have been handwaved (textbooks do this with the classic "(Why?)" inserted mid paragraph).
2:35-3:50 I think this is good; perhaps the idea of extending Q by the roots of any arbitrary polynomial was glossed over a little too quickly given how central the concept ends up being to the rest of the video (and the topic generally).
3:51-5:33 This part is fine, but it feels a little unclear as to what purpose it is serving in the overall video. I imagine it's trying to grow some intuition about how finite cyclic groups work when your elements are functions wrt repeated compositions, but this only feels like it is showing this connection as someone who has already seen it. It is not super clear to me whether an uninitiated student would be growing this intuition by watching this section.
5:34-5:50 Alright this is probably the biggest handwave of the entire video. I think building up the notion of what exactly the "symmetries" of an equation means is quite involved, and is not accomplished just by looking at the sqrt(2) -> -sqrt(2) example. In your defense, I think many textbooks also use this example and pretty much only this example, but it really is too complicated a concept to glean from just this example. There is a lot about field extensions and automorphisms that is being omitted here, and the viewer probably should be aware of this omission. Also, the notion of a "group" is kind of just introduced without any definition.
5:51-6:40 So, as other commenters have mentioned, this is actually not accurate. One natural question an attentive student might have is, "Aren't there 7!=5040 ways to map the roots of this equation to each other? Why do we only care about the transformations that take 1->2, 2->3, etc.?" This also gets a little more muddled since we are extending Q by the 7th root of 2, in addition to a primitive 7th root of unity. Using f(x)=x^7-1 here was probably better.
6:41-7:40 I think this is good. Minor nit: I think the numbers on the dial should probably be filled in with white or something, it can be a bit hard to read sometimes with the lighting.
7:41-8:17 I like the combination lock analogy, but technically the galois groups of these are incorrect. x^7-1 has a galois group of Z_6 and x^5-1 Z_4. Z_7 x Z_5 would have still been cyclic, btw.
8:18-10:19 I think this is okay, but this is one of the areas where you are probably shifting audience levels. Knowing that function composition doesn't always commute is pretty standard for math majors, perhaps not obvious for high schoolers and should be known to at least a good amount of STEM undergraduates. Prior to this point the video seemed good for all three audiences, but here it's appealing a bit much to one demographic and perhaps spending a disproportionate amount of time on it. In general I do like the clothes-wearing analogy for function commutativity. I think personally I would have liked this point in the video to justify why exactly these automorphisms (ie symmetries) commute rather than learning about what commutativity is.
The rest of the video is fine apart from the galois group computation errors, I think the transition at 11:54 is a little awkward since you go through an example where a group isn't abelian and then talk about groups where you can construct them from cyclic extensions, which are necessarily abelian, without any word like "however" or "on the other hand" so it feels like you're talking about the same thing.
The chaining of combination locks was good, and I think it captures the notion of direct products of cyclic groups well.
Other minor nit is that the music is not loud enough to add much to the video but probably not quiet enough to be totally ignored.
Overall I think the video is good; since it's your first, it's natural that there will be some feedback. At the end of the day I'm just a random dude with some feedback. My algebra is not super strong so I may have made mistakes in this comment as well. I hope you keep the spirit of this video and continue to make more, looking forward to seeing what your channel provides! :)

[jwvdvuurst]

And then the genius Galois thought it was necessary to have a duel, and died.

[mueezadam]

On YouTube, I’d rank you as one of the Top 3 explainers of dense mathematics. Galois theory was always presented as too abstract for beginner students yet this video gave me a good grasp of the basic tools this discipline offers.

I look forward to watching more of your content!

[lordipie]

it would be nice to see a video that explains why solvability of the Galois group is necessary and/or sufficient for the solvability by radicals, that would count as "Galois theory explained"

[shinli]

I like your idea very much, but I think there are things that can be improved.
6:15 Can have some explanation on why we can't shuffle the roots arbitrarily.
8:12 This is a cyclic group!
9:00 The wrong order.
11:21 May also try to visualize it as two dials (in addition to what you already have)

[roberthuber]

Probably one of the most beautiful fields of mathematics I have come across... everything from the content of the field itself and how a single teenager needed nothing but a simple problem to completely revolutionize our understanding of the world. I am so grateful to study such content in the coming months... when people ask me what math and physics is like I tell them it is stranger than you can ever imagine.

[sss]

6:23 I'm afraid that the Galois group of X^7-2 consist of 42 elements, not 7 (I believe it is equal to the semi direct product of the cyclic C_7 and C_6) . Apart from that, great video!

[mrsbrdvd]

I really liked the cosmic background music you put in here. I am sure Galois also had such a trip before the day he died when he was waiting for sun to come up and writing his proof. He had the same sparks and clashes in his mind that he felt that it was necessary that although nobody listened to what he needed to say, it was important that he expressed himself. He says in his notes,
"sun is almost rising, I have to hurry up...."

[aplacefaraway]

the cyclic visualization is really helpful.

[clementdato]

Great video! 9:26 we have a minor mistake: \phi\circ\lambda is to first apply \lambda then \phi, but the audio takes it in the opposite direction. Hope this help!

[xyzct]

Clear and simple. Thank you. It's is so much easier to dig in deeper when one has a clear overview like this.

[marcgoossens]

Just great. Took algebra (fields and groups) 40+ years ago: this was a pleasant refresher. And I like your general statement on swapping the study of an object for that of its symmetries: it's also what you do with symmetry groups in physics all the time.

[rtravkin]

Thank you for popularizing Galois theory! However, in addition to the mistake pointed out in the description, I think there's a mistake around 7:19 : the Galois group of the equation x^7 – 2 = 0 over the rationals is not cyclic but rather an extension of the cyclic group of order 6 permuting primitive 7th roots of unity by the cyclic group of order 7 that acts on 7th roots of 2 by multiplication by 7th roots of 1.

[delyank]

Nice video. I like the "military maneuver" metaphor. People like moving from more "analytic" concepts (e.g., how a polynomial function behaves) to more "algebraic" terms (like groups here). The ideal algebraic thing would of course have been to obtain a general expression (or algorithm) for all the roots, but unfortunately we cannot fully win that "war".

[legendddhgf]

Really liking how this channel presents stuff. Thanks for making this content, I am trying to self-teach math since I don't want to go back to school for a math degree.

[dcterr]

Excellent video. You made an inherently complicated subject comprehensible by clear explanations and clever use of graphics - well done! I look forward to watching some of your other math videos.