Galois Theory Explained Simply

To learn more about various areas of Group Theory: en.wikipedia.org/wiki/Group_t...
Galois Theory article in Encyclopedia of Mathematics: encyclopediaofmath.org/wiki/G....
[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.]
Some ideas in this video where inspired by Dan Goodman’s article “An Introduction to Galois Theory”: nrich.maths.org/1422.
The “trousers” metaphor was based on similar analogy provided by Eric Weinstein in one of Lex Friedman’s podcasts. (This and many other interesting episodes can be found at: / lexfridman )
To learn more about mathematicians that made a substantial contribution to Galois Theory:
Evariste Galois (en.wikipedia.org/wiki/%C3%89v...)
Niels Abel (en.wikipedia.org/wiki/Niels_H...)
Joseph-Lois Lagrange (en.wikipedia.org/wiki/Joseph-...)
Paolo Ruffini (en.wikipedia.org/wiki/Paolo_R...)
Do not stop exploring!

[@hannesk2119]

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!

[@donotlike4anonymus594]

A bit slow in my opinion but yeah... (The KZbin algorithm really seems to've liked this video compared to other math ones....)

[@AzureSky6612]

+1

[@MenkoDany]

I'm sorry but what the fuck how do you only have one video???

[@HansPeterSloot]

To me it is the first time I got an inkling of what abstract algebra actually is.

[@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.

[@yash1152]

i wish this comment rises to further top

[@luyombojonathan6688]

This comment is true

[@pendragon7600]

also a 2d reflection is not equivalent to a 180° rotation

[@bernardmarquot996]

I overall agree with your quite useful comment, but the Galois group G of x^7-2 is not dihedral with order 14. Indeed, the splitting field of x^7-2 is generated by the (real) 7th root of 2, along with the (all complex, except 1 of course) 7th roots of unity, whose Galois group is (Z/7Z)*=Z/6Z=C6. The degree of the extension (and so the order of the G) is therefore 7*6=42. Moreover, if you call r2 the real 7th-root of 2 and z=exp(2ipi/7), then Q(r2,z)/Q(z) is Galois with degree 7, so has Galois group C7. As the intermediate extension Q(z)/Q is Galois, the subgroup C7 is normal in G=Gal(Q(r2,z)/Q), with factor group Gal(Q(z)/Q)=(Z/7Z)*=C6. Now, by the Schur-Zassenhaus theorem, as 6 and 7 are relatively prime, the group G splits as a semidirect product of C7 by C6. In particular, it cannot by dihedral. By the way, the same arguments apply for polynomials of the form x^p-n where p is an odd prime and n is an integer different from -1,0,1: its Galois group is a semi-direct product of Cp by C(p-1), the latter acting on Cp through the isomorphism C(p-1)=Aut(Cp). [sorry for the long comment]

[@maynardtrendle820]

I came looking for just such a set of comments. 🙂

[@jwvdvuurst]

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

[@lawrencedoliveiro9104]

Imagine if he’d thought, “Bugger it, I’ll write this up tomorrow night. I’m going to bed.” ...

[@TALKmd]

Exactly! , this was too bad we lost also in this dual of his .

[@joshwalker7460]

I want to live in the universe where Galois won the duel and survived.

[@ga35am]

Well, if the École Polytechnique's examiners weren't so shitty and his father hadn't comitted suicide, maybe Galois wouldn't have gone to that duel. Anyway, it was definitely NOT just a matter of thinking that a duel was necessary and dying. While Galois was probably a genius, a bunch of mediocre people have a lot more shameful or questionable behaviors or words than dying in a duel and still are recognised are geniuses. Dying in a duel is actually very honorable.

[@headlibrarian1996]

Galois was a hothead.

[@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.

[@whatelseison8970]

That doesn't have much to do with radicals though. If anything it's a coincidence related to auditory perception of frequency being logarithmic. Sure 440 Hz and 880 Hz are A4 and A5 but they are two different numbers and two different notes despite being harmonically similar. In the mathematical sense its the 12 complex 12th roots of 2 that form the cyclic group but when considered as complex frequencies they actually correspond to the phase of the oscillation, not the frequency. Hence why they can come back to where they start in a closed cycle. I hope that makes sense. I'm having a hard time explaining this as clearly as I'd like to.

[@amankarunakaran6346]

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 3*sqrt(2)+(4-sqrt(2))/(5+sqrt(2)). 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! :)

[@mrtaurho8846]

It's very good that you commented on some of the more severe mistakes in this video. It appears that the creator confused the Galois group of a field extension and of a polynomial equation (which may differ, when only adjoining one root and not using, say, primitive elements) at some points, or rather didn't check his computations. There is a relevant post on reddit ( www.reddit.com/r/math/comments/kk7cde/galois_theory_explained_visually_the_best/? ) also talking about the problems. (Note: I really loved the video but I think the number of mistakes is problematic)

[@pauls.2451]

I am doing an MSc. in maths and found this video fantastic. Certainly won't make up for your maths course and hours of drilling exercises, but it's nice to see this stuff explained in normal non-convoluted language for once.

[@amankarunakaran6346]

​@@pauls.2451 I agree that it is nice to provide some "disillusionment" from how scary higher level math seems, but that kind of thing needs to be done with care. If you stray too far from rigor and too close to "wow" factors you end up with some problematic videos like numberphile's infamous disastrous -1/12 video, which has forever plagued the way many laypeople see math. Much of the best parts of this video build some intuition about cyclic groups and their direct products, but give a bit of incorrect intuition about what exactly Galois groups are. Don't get me wrong, I love the fact that someone is taking on the challenge of explaining Galois theory to laypeople, because I think this type of thing can really get people excited to see what math is really all about. But it requires a great deal of attention to detail to make sure that you are really conveying what should be conveyed, rather than what people will digest easily at the expense of precision. Btw, if you or anyone else is looking for a layman friendly book to get into the subject, I've worked through quite a bit of "Field Theory and its Classical Problems" by Hadlock and it's really quite good. No prior algebra knowledge assumed.

[@Tadesan]

wrt? With regards to? Honestly, if you to write a thousand word youtub comment and decide to be cute by abbreviating three words; I kind of don’t respect you. Jmoymmv.

[@192ali1]

Dear Aman, Assalamu Alaikum, in case you are a Muslim and "Greetings," in case you are not. You seem to know so much about the Galois theory. Why do you not make some of your own videos. I hope I can have email correspondence with you and ask questions about this subject. I cannot give my e address explicitly. it will be erased so: ali.jamily1at g.come. thank you

[@pasfier7098]

Bruh. This explained everything SO MUCH BETTER. Wish the youtube algorithm showed me this earlier.

[@matferro]

To be fair yt algorithm showed us this video with only one month delay. Usually it shows me the videos I want to see with 8 to 10 years of waiting. This time it did a great job

[@Rahul-uk4su]

Bro the channel is just a month old chill

[@Tadesan]

Bril. Don’t say bruh. It makes me feel that I’m listening to an idiot.

[@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"

[@mimikal7548]

Yeah I still don't know why this is true after watching this video. It seems that this is the most important part of the theory so it's strange not to explain it.

[@dylancrooks6548]

@@mimikal7548 obviously this comment is late, it is more an exercise in me trying to explain it. I think that, at least for sufficiency, the groups must be cyclic, so that when you apply more groups, you won't continuously be making algebraic equivalences that get you farther and farther away from the solution. it's like going into a funnel vs coming up from a funnel, it is easy one way and hard the other. (do not read this part, unless you have a better explanation - in review, this is a horrible and confusing explanation) It might be possible to think about it this way: all the information is contained within the equation and within the rules of mathematics. we can view symmetries as changing context. Consider a jigsaw puzzle, where you have the puzzle (we'll say it's like a line, in that there's one place to put a piece and a finite amount of jigsaw puzzles left). When you place a piece and it fits (let's say that's symmetric), you'll obviously see the jigsaw puzzle and say, yes, that's right, just like you can sometimes tell when solving a polynomial equation that you're getting off track and terms are growing larger and such. When you apply another symmetry, you preserve the "nature" of the equation, but change how it looks - just like when doing algebra, your 5th step is equivalent to your 1st step, because you do everything to both sides of the equal sign. So, if the equation can be seen as a galois group with cyclic component groups, then obviously, there is a solution. It's like backtracking through a maze, it is easier from the end to the start than start to end (NP vs P). By tracking the grammar of the math, while whatever format the problem is in might change, Galois theory can tell us if the overall problem has a solution, or if it doesn't. Which is pretty cool, it is like framing a painting in different frames. The style of the painting and the effect it has on your room might change, but the painting stays the same. Or, it is like learning. The subject you learn stays the same, you just need to change how you see it, until you finally grasp it. Obviously, changing how you see it is done automatically, by your brain, although you can see it at a psychological level if you want.

[@sss29489]

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!

[@fabriziomori1128]

Yes, it's true.

[@JustinLe]

probably a typo of x^7-1 :'(

[@fabriziomori1128]

@@JustinLe no because in that case is an extension of degree 6 lol. By the way it's a good video, no one really noticed it 😆😆

[@David-km2ie]

If the roots of unity are included in the base field its still right. Which I assume since that is what ables us to use rotations.

[@fabriziomori1128]

@@David-km2ie yes i was assuming the base field was Q obviously! But whatever.. if everyone understood the right context shame on me 🙈

[@meghanabhange13]

This is one of the best explanations of Galois theory I’ve seen. I’m physically exhausted by how many people I’ve shared it with my friends. It’s so intuitive that now I finally have a place to redirect people who are scared of Field Theory

[@mueezadam8438]

On KZbin, 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!

[@aiwen6942]

I'd be curious to know what the other two top channels are in your ranking :)

[@mueezadam8438]

@@aiwen6942 Aleph Null and of course, Grant Sanderson!

[@aiwen6942]

@@mueezadam8438 Cool - thanks! I came across Aleph 0 very recently and then the KZbin algo also recommended this video. It's good to see these newer channels begin to rise up to the very high bar set by 3Blue1Brown

[@shinli256]

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)

[@shinli256]

@@__mrmino__ That's exactly what's missing. Galois groups are the maps that preserve addition and multiplication (as in the first example). Naturally these maps are permutations of roots, plus it's sufficient and easier to consider the roots. However just a random permutation might not be obtained from a member of the Galois group.

[@aplacefaraway]

the cyclic visualization is really helpful.

[@dcterr1]

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.

[@xyzct]

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

[@clementdato6328]

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!

[@Miyelsh]

Often more introductory explanations of group theory use left-to-right instead of right-to-left. There are a few group theory lectures on youtube that I've seen the same.

[@brianb2308]

It definitely depends on which book you use, but yes, usually the transformation is applied right-to-left because it's like functions sorta.

[@roberthuber2770]

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.

[@yoannmery]

I've been looking for this video for a long time. You managed to keep all the juice with the right amount of definition. Thanks!!

[@marcgoossens1658]

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.

[@milisicht]

Studying this stuff in uni, obviously in greater details, but this gave me a better perspective on some things like cyclic groups. It’s awesome when a video that is understandable for someone who does not know the subject still helps complete the work of books and professors. Absolutely loved it!

[@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.

[@eragon_argetlam]

Am I the only non math enthusiast here who had no idea what he was talking about half the time, but still watched cause math's interesting as fork?

[@revimfadli4666]

More interesting than spork, even

[@ahmednesartahsinchoudhury2628]

sem

[@user-yk1lz7gb2t]

It's even better than sex 😂

[@HansBaier]

Excellent! Understanding the essence of Galois Theory in 15 minutes. Worth every second!

[@doodelay]

This is becoming my favorite maths channel please make more videos!

[@mariorqmsilveira3270]

Really good class. Build up from simplicity and comprehensive examples. Love it!

[@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.

[@orangeguy5463]

Yeah he didn't really describe how the galois group is obtained at all. It is assumed to preserve multiplication of numbers in the field and built up from there. Then in simpler terms if you have 7 roots of 2, the quotient of any two is a complex 7th root of 1, and these roots of unity must permute among themselves as elements in this field, in a way that fixes the trivial root 1. The one with the smallest complex argument (angle from positive real) generates the rest and is called primitive. Then a permutation in the galois group for x^7-2 is given by finding where this primitive root of 1 goes, and where the real 7th root of 2 goes. But it's still solvable as a group because, without too much group theory, we can work with the cyclic subgroup that fixes the primitive root and only permutes the roots of x^7-2 by multiplying by some fixed complex root 7th root of 1. A good way to think about it is that the naive way you'd want to permute the 7 roots of 2 is with a group having 7!= 5040 elements, highly nonabelian (though still solvable, that's besides the point). This roots of unity business is reducing the complexity to a small subgroup and proves solvability, it shouldn't be thought of as making it harder.

[@rtravkin]

@@orangeguy5463 That the group S_7 of permutations of 7 elements is solvable is false! General degree n equation isn't solvable in radicals for n ≥ 5 precisely because S_n isn't solvable.

[@christopherellis2663]

x⁷-2=0 x⁷=2 x=2^(¹÷⁷)

[@samsonblack]

This is a lovely video. In fact, I don't think I've ever seen solvability by radicals explained so clearly and concisely. Thank you!

[@resonancyone]

As a Ph.D. in Math, I didn't even know this! Thanks!

[@Jooolse]

Oups, this is typically studied at Bachelor level ;-)

[@resonancyone]

@@Jooolse well, yeah, keep learning! lol

[@IsomerSoma]

@@Jooolse depends what modules you choose and what uni you study at. You could ignore algebra courses and focus on Analysis instead.

[@mrsbrdvd6955]

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...."

[@michaelzumpano7318]

This was a great intro to Galois theory. Different and better than I’ve seen before. I hope you’ll do a whole series. I’d love to see how these concepts evolve into Lie groups and to solutions of physical problems.

[@melanthajezek2835]

I love this, thank you so much for clearing up all doubts I had about studying Group Theory. You've rightfully earned a subscriber!

[@joyboricua3721]

I've heard of this topic before but your approach in explaining with such visualization is very well crafted. Thanks

[@Sidionian]

Nice. Keep up the good work. Algebraic Topology and Algebraic Geometry concepts away you, my friend.

[@lujoconnor]

Galois theory was one of my very favorite units what must be 10 years ago now. Thank you for this beautiful video - a fun tour down memory lane

[@pectenmaximus231]

This is a very, very beautiful video. This is how you spoonfeed and it’s wonderful of you to have put this together, thank you.

[@jongraham7362]

I may never understand Galois Groups but this video has already helped me get closer than I have ever gotten to understanding them.

[@doctormcgoveran2194]

Loved the video. It ties together so many of the ideas I had floating around loose.

[@theboombody]

As far as I can tell this is the only video this channel has so far, but it sure started out with a bang. I'm glad people post this kind of stuff on youtube because a lot of the literature out there is so much less accessible. Because of channels like this, I can eventually see a future where one day Galois theory will be just as accessible as calculus is today. Still challenging, but accessible. Maybe even at the high school level.

[@scotthader705]

Thanks so much for putting this so simply. Now I kind of get the motivation of using a derived series in the definition for solvable groups.

[@yatima1158]

Great video! Loved the dial visual of cyclic group extensions.

[@lawrencedoliveiro9104]

9:48 Superman needs to see this.

[@timothebillod-morel6777]

Very good work sir, I work every day with Galois groupes, if I had to explain it to someone that doesn't know maths, I would do something like that.

[@carlosalbertocuadros5469]

simple and direct , easy but interesting subject EXCELLENT JOB.

[@glarynth]

I'm getting in on the ground floor. Looking forward to your 100k-subscriber special!

[@pdelong42]

Thank you for this explanation. I've been studying group theory on and off for years, but I always stop short of diving into Galois theory, because it seems difficult to approach. But this gives me the motivation I was looking for.

[@titusfx]

Looking for more videos... was really disappointed to see only one video 😅. Excellent video!!! Thank you! 👏👏👏

[@m322_yt]

Very interesting, cleared up a lot of question marks concerning motivation left after taking a Galois theory class

[@aaronTNGDS9]

Excellent presentation, great pace and animation.

[@delyank0]

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".

[@mannyc6649]

I disagree, I think it's much cooler that we know that there are no solutions for n>4, it's one of the most surprising facts in math. If there was an algorithm for every polynomial, first, chances are that it would be so ugly that nobody would write it down for all n, second, it would be more convenient to solve a polynomial via Netwton method if one needs to know the root numerically.

[@strangeWaters]

Question: is Newton's method guaranteed to converge? Do we have a proof that it "solves" all polynomials, in some approximate sense?

[@mannyc6649]

@@strangeWaters You need to be "close enough" to the root. So you kinda have to plot the polynomial first to see roughly where the roots are.

[@delyank0]

@@strangeWaters Yes, there are typical textbook proofs that show that under the assumptions of smoothness (always the case for polynomials) and being "sufficiently close" to the root, Newton's method will converge. For simple roots, once being sufficiently close to the root, the convergence is quadratic, which is a very nice property of Newton's method. For multiple roots the situation is more complicated in practice. While standard fixed-point theory would still provide geometric convergence, the actual rate worsens as the multiplicity increases. So some special care is needed. Apart from these mathematical consequences of the multiplicity of roots (which hold even under perfect arithmetic), there are other numerical aspects. A multiple root suffers from ill-conditioning in the sense that there is larger margin of error due to amplified effects of round-off noise and the ways you evaluate the polynomial may affect the error more. After all, there are some reasons why people don't prefer polynomial root finding approaches even for things that seem natural candidates for that, like finding the eigenvalues of a matrix. In fact, in some cases (like orthogonal polynomials) one would opt for reformulating the root-finding problem as one of finding eigenvalues of a matrix. Interestingly, for complex polynomials, if you color the points in the complex plane according to which root is recovered by the Newton's method starting from that point and how many iteration were necessary (or if the method failed to converge), you end up plotting some fractals. Newton's method has intrigued a lot of the greatest minds, Kantorovich being one of them.

[@xaviermachiavelli5236]

@@strangeWaters SO, B~₩◇▪︎•○OS a.

[@sali-math-arts2769]

Very nice and understandable explanation of the link between solvable polynomials and solvable Galois-Groups of polynomials. Thank you so much for showing the core ideas

[@Spencero123]

I think there is a note of confusion during the segment on composition of transformations. If one maps an element of a set through a composition of transformations, then the transformation on the right side will be applied to the element first. During the segment where transformations were demonstrated via the wearing of clothing, the left transformation was applied first. However, this video was very helpful to me, and I am looking forward to more content! Thank you.

[@lakshaymd]

Great video, but some examples are wrong. The Galois group of x⁷ - 2 is not C₇. The degree of the splitting field over Q would be 42, as adjoining the real seventh root of 2 gives you a subfield of the reals, after which you need to adjoin a primitive 7th root of unity, whose minimal polynomial over Q(⁷√2) is the same as the one over Q using the tower law. The Galois group is actually a semi direct product of C₆ and C₇. If you change the polynomial to x⁷-1, you get C₆ as the Galois group. The Galois group of (x⁷-1)(x⁵-1) is not a direct product of C₅ and C₇. The degree of its splitting field is (I think) 24 (adjoin a primitive 5th root of unity then a 7th root, showing that the minimal polynomial of the 7th root over the intermediate extension has degree 6 might need some work idk). If it is 24, then the Galois group is the direct product of C₄ and C₆.

[@brymermeneses4975]

I’m just dumb senior in HS with little to zero knowledge in abstract algebra (I tried to study it myself but I was not able to grasp the abstraction) this gave me a vague sense about galois theory and motivated me to continue studying it!

[@zeldamage001]

Awesome video, very clear! You didn't go into much depth, but you hinted at enough terms and theorems to allow one to five deeper based on the video. Very nice!

[@JM-us3fr]

This was a terrific introduction. Great job

[@PhilipSmolen]

I am anxiously awaiting part II.

[@richlizard709]

R u single

[@joshwalker7460]

This was incredibly helpful, thank you.

[@borjavazquez264]

Matrix/vector operations visualized like this would be amazing, such as projections, dot-products...

[@bookashkin]

Look at 3Blue1Brown Essence of Linear Algebra videos.

[@surfing_youtube]

I really liked this video. You are doing a great job. I heard about Galois theory from many people but never actually knew what it is. This video explained everything in a nutshell.

[@user-qr8yv7ys4e]

Great explanation, you are going to be big on youtube one day!

[@trueriver1950]

Great! A clear explanation of what Galois theory aims to prove. It would be great if courses on this started with this overview, to give an idea of where they are going

[@nishapasricha427]

Really helpful and amazing video . keep making more visual video

[@poirotpotiron6307]

Thank you ! Your explanations are so clear !

[@cowgomoo444]

Wow I have little understanding of math past a high school level but I was able to understand(more or less) what you were presenting. You have a gift for presentation I hope you continue to make videos.

[@g3452sgp]

This is a really great video on Galois group theory!

[@Fetrose]

Wow. What a great beautiful presentation. I learned a lot. Thank you so much and keep going.

[@Samfhire]

The Galois group for x^7-2 isn’t the cyclic group with 6 or 7 elements, like it says in the video. It is for x^7-1 but not x^7-2. The rotation symmetry where the seventh root of 2 gets multiplied by the seventh root of unity does generate a cyclic group but there are other symmetries too. Let r be a primitive seventh root of unity, then r can get mapped to r^2 (and r^2 goes to r^4 etc.). That generates another cyclic group with 6 elements and these two cyclic groups combine together to give another group with 42 elements. You could write the presentation for this Galois group as I’m not sure what that group would be called though.

[@gresach]

Yes I agree f(x) = x^7 − 2 The 7 zeros are 2^(1/7)* k^j where k is the 7th root of 1, and j=0,...,6. Splitting field: F = Q[2^(1/7),k]. The minimum polynomial of 2^(1/7) is x^7−2 So |Q[2^(1/7)]:Q| = 7. The minimum polynomial for k over Q is x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 & over Q[2^(1/7)] it is the same. |F:Q| = |F:Q[2^(1/7)]| * |Q[2^(1/7)]:Q| = 6 * 7 = 42. The Galois group has order 42. AMs can be. 2^(1/7) -> 2^(1/7)*k^j, for j=0,...,6, and k -> k^j, for j=1,...,6. Giving 42 combinations, which all happen. These two subgroups are C_7 & C_6 but do not commute.

[@meruscales]

Really enjoyed this. Have you thought about making a video about permutation groups and cyclic groups specifically? Maybe a proof of 5+ elements not being able to decompose into cyclic groups? That is a very fun thing to show visually

[@mzimmer1751]

Great job explaining this! Very clear

[@waterduck636]

This is gold. I already took field theory class but this vid really helped me understanding galois theory.

[@lolmomz]

Brilliantly explained! Subscribed.

[@DooDooDiaperShitCunt]

This was really fantastic! Thank you so much for this!

[@dr.soumenkanrar7254]

Very Good Presentation, Thanks.

[@maxsch.6555]

Absolutely fantastic work. Keep it up!

[@kquat7899]

Very nicely explained.

[@kharnakcrux2650]

after thinking back... this took me almost 2 decades to fully digest... But in my explorations with other things... like the Zeta function, Theta functions, fractional calculus... performing operations in an eerily... non-integer way.. i just took those leaps and tried out some odd ideas. i felt they were too odd or offbeat to be anything significant, but here it is... Hot damn i swear.... turns out things like Galois theory, combinatorics, and abstract maths, seem to have been there the whole time. like a kind of bridge between all these different branches of maths.

[@Hamza-zz3zc]

Very interesting video with a very deep and profound explanation .. thank you 👍

[@UxJoy]

Love the presentation style, I’ve always struggled with math because I’m more of a visual learner - would it make sense for you to do a basics series as well? :)

[@pinklady7184]

Same here.

[@yee9543]

I love the background music, it is so soothing that I felt like I was peacefully dying in sleep.

[@ilioscio]

Pretty darn good video, and I appreciate your links to extra reading materials in the video details. Sub'd.

[@matveyshishov]

This is the absolute best explanation, thank you :) !

[@iankeck3419]

Looking forward to further videos.

[@speadskater]

Yes, please do more, this is wonderful.

[@frederik3982]

great work, looking forward to more videos from your channel

[@harambe_andy5134]

This is fantastic! Thank you!

[@Debanil12]

Mind-blowing explanation.

[@newsgo1876]

I didn't realize the BGM at frist but soon I found what made me feel thrilled...

[@dyllanusher1379]

Loved the informative video!

[@AdobadoFantastico]

Good stuff, great visualization.

[@utubecorporatetroll]

Subscribed ! Looking forward to more videos from your channel.

[@GregBakker]

Nice video! Looking forward to your next one.

[@alexandersanchez9138]

Great stuff. Keep doing videos, bro.

[@Caturiya]

Very nicely explaind. I also studied mathematics, but neglecting much algebra. What you tell and the way you tell is simple and easily to be understood, but for me a non familiar with algebra not familiar. Books on Galois theory often ignore this NON FAMILIAR but you ignore not. THANKS FOR GOOD DIDACTIC

[@gregoryconnor3134]

Excellent discussion.

[@glendale3070]

Interesting video that explains a highly complicated subject in simple ways. Looking forward to a video that explains why the equation is solvable by radicals if and only if the corresponding group is solvable.

[@alfredoduran5149]

Hi, I have watched your video 5 times so far. It is refreshing and very very original. Thanks you for sharing this wonderful way to simplify the Galois Theory. I have been interested on the Quintic and the solvability of polynomials all my life. A lot of work to put all the essence on the theory in a 15 minutes video. You are a genius ! How did you do it ? What software do you use ? The result is amazing... Bravo ! Can you please put some more information about yourself ?

[@corneliusgoh]

Nice Video👍 very intuitive.

[@juanpgomez1299]

This passed WAY over my head

[@davidbacon9223]

spare change, bro?

[@FreestateofOkondor]

awesome explanation! please keep making videos

[@viggosimonsen]

Great presentation I would also have mentioned that by the same token, this is how the complex field is constructed - by adjoining i to R, corresponding to the equation X^2 +1 = 0