Grant Sanderson (3Blue1Brown) | Unsolvability of the Quintic | The Cartesian Cafe w/ Timothy Nguyen
ฝัง
- เผยแพร่เมื่อ 6 ก.พ. 2025
- Grant Sanderson is a mathematician who is the author of the TH-cam channel “3Blue1Brown”, viewed by millions for its beautiful blend of visual animation and mathematical pedagogy. His channel covers a wide range of mathematical topics, which to name a few include calculus, quaternions, epidemic modeling, and artificial neural networks. Grant received his bachelor's degree in mathematics from Stanford University and has worked with a variety of mathematics educators and outlets, including Khan Academy, The Art of Problem Solving, MIT OpenCourseWare, Numberphile, and Quanta Magazine.
In this episode, we discuss the famous unsolvability of quintic polynomials: there exists no formula, consisting only of finitely many arithmetic operations and radicals, for expressing the roots of a general fifth degree polynomial in terms of the polynomial's coefficients. The standard proof that is taught in abstract algebra courses uses the machinery of Galois theory. Instead of following that route, Grant and I proceed in barebones style along (somewhat) historical lines by first solving quadratics, cubics, and quartics. Along the way, we present the insights obtained by Lagrange that motivate a very natural combinatorial question, which contains the germs of modern group theory and Galois theory and whose answer suggests that the quintic is unsolvable (later confirmed through the work of Abel and Galois). We end with some informal discussions about Abel's proof and the topological proof due to Vladimir Arnold.
#3blue1brown #grantsanderson #math #maths #mathematics #algebra #grouptheory #pedagogy #equations #polynomials
Patreon: / timothynguyen
Part I. Introduction
00:00: Introduction
00:52: How did you get interested in math?
06:30: Future of math pedagogy and AI
12:03: Overview. How Grant got interested in unsolvability of the quintic
15:26: Problem formulation
17:42: History of solving polynomial equations
19:50: Po-Shen Loh
Part II. Working Up to the Quintic
28:06: Quadratics
34:38 : Cubics
37:20: Viete’s formulas
48:51: Math duels over solving cubics: del Ferro, Fiorre, Tartaglia, Cardano, Ferrari
53:24: Prose poetry of solving cubics
54:30: Cardano’s Formula derivation
1:03:22: Resolvent
1:04:10: Why exactly 3 roots from Cardano’s formula?
Part III. Thinking More Systematically
1:12:25: Takeaways and Lagrange’s insight into why quintic might be unsolvable
1:17:20: Origins of group theory?
1:23:29: History’s First Whiff of Galois Theory
1:25:24: Fundamental Theorem of Symmetric Polynomials
1:30:18: Solving the quartic from the resolvent
1:40:08: Recap of overall logic
Part IV. Unsolvability of the Quintic
1:52:30: S_5 and A_5 group actions
2:01:18: Lagrange’s approach fails!
2:04:01: Abel’s proof
2:06:16: Arnold’s Topological Proof
2:18:22: Closing Remarks
Further Reading on Arnold's Topological Proof of Unsolvability of the Quintic:
1) L. Goldmakher. web.williams.e...
2) B. Katz. • Short proof of Abel's ...
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![Revolutionary Math Proof No One Could Explain...Until Now [Part 1]](http://i.ytimg.com/vi/RX1tZv_Nv4Y/mqdefault.jpg)
I think Feynman said that the solution of the cubic was the most important development in mathematics because it proved that we could know more than the ancients - ie, that there could be progress.
Ok buddy
@@oldboy117?
And yet I have no clue how the cubic solution works.
Nobody said that *you* could know more than the ancients...
More seriously, there's a good Mathologer video about it iirc
Bless your calculus teacher, and everyone else like him or her - the ones who see potential in a child and make a point to encourage them. Those people are heroes.
This chat was lovely! The greatest lamentation in the history of mathematics is that Galois and Abel didn't each give us 40 more years of their genius.
Don’t forget Ramanujan
39:40 @Timothy Nguyen You are right. Once you have figured out the mean, you can actually manipulate the expression to make it a root of a related polynomial. So you can simply factor out one of the roots. What you are left with is a quadratic expression that will give you the other two roots. Suppose the original equation is ax^3+bx^2+cx+d=0. Now, mean = -b/3a. Solve for d_0 from the equation mean = (-b^3+27a^2d_0)/(9ab^2-27a^2c). Now you can replace d in the original equation with d_0. x=-b/3a is now one of the roots of the transformed equation. All we are doing is shifting the original equation up or down such that the point of symmetry becomes one of the solutions. By this transformation all the 3 roots have changed. But we can recover the original roots by solving y=delta with the new transformed equation. delta = d_0-d.
I actually computed the roots by first linearly approximating the transformed quadratic equation about x=-b/3a. y=f'(-b/3a)x + f(-b/3a). Solve it for x putting y=delta. Then you can linearize the function again about the new value of x. Solve it with y=delta. Repeating this process you can get approximate roots with as much accuracy as you may want. But this is kind of a nice tangent to explore. Your topic is about insolvability in terms of radical expression.
This format is awesome, you really get into the meat and bones of the topics and learn so much more than just popular level STEM communication.
Wow Tim - really impressed with the guest list you have on this channel. And now add Grant to it!
This channel is going to blow up. Excited to see it and well deserved! 👌
(Haven’t listened to it yet, but I did just find my listening material for my run later today)
45:46 - Because we're not talking about emotions. It's an entirely unrelated meaning and we don't even need to bring up emotional depression.
i agree
its a maths podcast what use is there in bringing this up??
Tim, please keep up your fantastic work, and bringing Grant on is hopefully the new addition that you will revisit on many other important topics. I learned a lot from this one and stepping through history is another way of motivating the intuition for the modern versions of these core concepts and how to problem solve and the mechanics involved in mathematical insight and intuition in general. I picked up one or two new concepts that I'm going to try to apply to my own research. Thanks a bunch!
Hey, I really like this episode! i like how Grant and you present the origins of group theory in an informal sketchy way, it's improvised but never without authentic opinion and flavored with your own characters, definitely more tasty than text book thank you!
Your mood is incredible, thank you for the video
As someone struggling with depression, I love the idea of a depressed polynomial, especially because it's one that's "staying at home" (it's shifted to be around x=0) and needs to get out there more to stop being depressed.
this is the one I had enjoyed the most! Thank you
Hi Tim! Great to see you still doing awesome things with awesome people :)
what a coincidence! im doing an undergraduate thesis on galois theory because im interested in the origins of groups as well! im also attempting to make sense of how groups manifest when solving solvable polynomials such as in Dummit's paper over solvable quintics.
Man I love working through these with you guys. It's like working on some problems with my friends.
This was truly enlightening. I wish I'd been taught it like this.
At 1:49:36 Grant makes a great point about this topic: this stuff is complicated, but we're also asking a fairly contrived question, "how can we express solutions in terms of radicals." It's really cool that we found when and how it's possible, but the techniques are so much more important than the solutions by radicals themselves, which are generally a pretty terrible representation of a solution to an equation... for practical or even theoretical purposes!
The deep importance of Galois theory, just on its own without group theory as a retroactive motivator, goes way beyond just insolubility of quintic and it lies in what can be implied about the structure of our “solution space” by looking at the group of permutations of this “solution space”. As in much of the big algebraic fields, algebraic number theory and algebraic geometry, have to imply some structure about their solution spaces by looking at something simpler, most famously supposing their exists integer solutions to fermats last theorem implies a solution space exists with a specific structure that when that structure is looked at more generally contradicts something about the symmetry of such a structure. That symmetry being understood by the representations of a type of galois group! Galois theory is taught not just because it kind of feels like the most obvious first use case of actual group theory, but because the tools it builds up are still useful in the most modern methods of math today yet doesn’t require the same level of background as algebraic topology or lie theory. This gets much, much deeper than this compared to some of the stuff like class groups and invariant theory which are still important but feel minuscule compared to the massive ways that Galois theory has infested so many fields.
Beautiful. Thanks a lot, gentlemen.
CompSci graduate here. I was always intrigued by polynomials, their nature and solvability. I am halfway into this converation and I can already say that it is a gem. First - because of the guest and how the conversation is conducted, second - the insights and the thought process along the solvability of polynomials of successive degrees are pure amazing. Thank you both.
PS Let me bring in my two cents. I stumbled upon Sylvester matrices sometime ago. I just thought to myself if there existed a viable way to 'probe' whether a polynomial of, say, degree 5 had real roots by building a Sylvester matrix with its coefficients and coefficients of lower-degree polynomial of known roots and testing the determinant of that matrix to reason whether a polynomial like that can be effectively 'guess-reduced' to, say, a linear times quartic or quadratic times cubic that can be easily solved. I am not sure it that makes sense, but I just post this idea for you to explore as well (maybe it is a no-brainer in the math community, I don't know - just tossing an idea).
Years ago I saw your video on quantum YM in 2D, now I chanced upon the same channel! Subscribed!!
This is work of Neils Abel. Maybe we can also create a clip on Taniyama-Shimura conjecture, as an example of more recent breakthroughs in number theory (i.e. Galois representations)?
It's amazing how prolific Arnold was. I just bought his mechanics book, based on your talk with Baez. My PhD was a long time ago, on singularity theory, which is an area basically discovered by Arnold. (It was pure maths, nothing to do with black holes or Kurzweil.)
Great video, but 1,2,4 are not the roots of x^3 - 7x^2 + 11x - 8. Guessing and checking at about 38:50 was careless; indeed 1*2 + 2*4 + 1*4 = 14 and not 11. Maybe Grant meant to write 14x...
I enjoyed the first segment about AI and its' potential to impact mathematics. I think once AI is in a place where it is genuinely replacing large parts of what mathematicians do, i.e. proving theorems with cohesive solutions, I can't see why AI wouldn't be able to do so much else that will replace many other activities we consider intrinsically human.
I am a software engineer and many have panicked over the potential for AI, like Github Copilot, to replace software engineers. I cannot really see how a tool like Copilot could replace any serious software engineer. I think it's hard for people outside of the field to imagine what it's like, but my best explanation is it's like being a detective, an engineer, and an artist all in one (but usually failing to excel at any of those things). For instance, if I am solving some production bug, this requires me not only understanding the code paths that are involved(this is actually the easiest part and usually comes last) but the services, the contracts between them, the data models for persistent storage, the protocol for transmitting data, etc. Much of this is not explicitly programmed.
If I am building a new service, I need to consider scale, existing services, legacy architecture, etc. If an AI can replace that, I don't see why an AI cannot replace anything, meaning we've essentially arrived at general AI.
In the next 10-30 years, I think AI in programming will become like AI in writing, where it serves as a tool to make a programmer more productive rather than anything like a replacement. I think this will be true for AI in many fields.
Loved this video! Lots of things clicked for me, and I enjoyed all the historical tangents :).
Thank you for the excellent content- very informative and well formatted!
wow internet what are you doing changing the world forever?
I think the note on the bottom-right of 1:18:45 may be confusing because it uses the • symbol in an unfamiliar way. If g is a permutation of variables and p is a polynomial in those variables, g•p refers to the result of permuting the variables of p according to g.
Alas, I didn't define the group action in the note since it was implicit in the preceding discussion. Hope that doesn't cause confusion!
@@TimothyNguyen Sure. I just worry people might confuse it for multiplication.
Great first part
Second part was confusing, when you start handwaving through concepts and "5-cycles" "2-cycles" "symmetries" etc.,then you've become what you are trying to avoid!
But the answer is that the demand for solutions in terms of STANDARD radicals is too restrictive. However, if you generalise the notion of radicals, then the general quintic can be solved. In fact, the general quintic can be solved in terms of so-called 'Bring radicals', which are solutions of the special quintic x^5+x+a=0. The latter cannot be solved in terms of standard radicals, but if you add the real solution as an operation on a , then any quintic can be solved in terms of this new radical.
Thank you for pointing this out! Didn’t know this at all.
That's abit of a puzzle because if a = 2
x ^5 + x + 2 = 0 has real solution
x = -1
According to Wolfram Alpha if a = 3 or 7 for example it's not solvable by radicals.
Take any real or complex number it can be written down in a finite way using Natural numbers 1 2 3... etc and using add , subtract, multiply , division and taking roots OR IT CANNOT.
It's so clear, so no problem with Bing Radicals but they are not radicals in the usual sense or am I missing something. I do understand that there are other methods of solving quintics but numbers are radically expressible OR THEY ARE NOT.
My question is why are they called ultra radicals?
Please let me know I'm very interested - thank you
I do understand that we think in terms of a formula in terms of coefficients because obviously the coefficients uniquely determine the equation under investigation.
Generally and it is the SAME THING a number is radically expressible OR IT IS NOT , let's not consider if it's the root of some polynomial.
So question is, why are these other numbers called Ultra Radicals?
Another question I wish to ask is,
How can you generalize the usual definition of Radical?
You have the Natural numbers, the four elementary operations and roots, finite expression.
How is that generalized?
It's not the mathematics I'm questioning but the way people are thinking about it and the terminology being used.
A number is radically expressible OR IT IS NOT.
10:11:00 Some where around here you ask about redundancies. You can give perfectly clear explanations of this that makes it obvious. (Not that I'm helping anyone by just saying so).
If 1,2,4 are the roots of x^3 - 7x^2 + 11x - 8, how would their pairwise products sum to 11? It would have to be divisible by 2 to be plausible...
Exactly! I think he made a mistake.
I can't believe that Grant doesn't have a PhD. I
His work is pretty impressive.
Two of the dumbest people I've met have a PhD -- 1/10 on creativity and 10/10 for robot behavior. I don't think there is correlation between outdated education system credentials and intelligence.
😂
Keep every permutation once you make it from every step and no step can be intermingled with pervious steps… then you can get a more full way of computation in ways of degrees of freedom
47:27 - Oh, definitely go with general symbols. Much more informative.
I'm still looking for a "simple" proof of the non-existentence of a quintic formula, I'm guessing that the maths involved is rather esoteric. I'd really love to see a 3b1b-style video on it
SAAAME. I just want an explanation that I can see and get intuitively.
This all reminds me of a puzzle.
That reminds me of a puzzle: suppose we have access to _cyclically_ ordered tuples, so that
(a,b)=(b,a),
(a,b,c)=(b,c,a) (but doesn't equal (a,c,b)),
etc.
For which n can we construct unordered n-tuples?
(The analogy here is to think about how Kuratowski constructed an ordered pair out of unordered sets.)
Spoiler:
.
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:
:
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:
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n=2: (a,b)
n=3: ((a,b,c),(a,c,b))
n=4: Apply the solution for n=3 to ((a,b),(c,d)) and ((a,c),(b,d)) and ((a,d),(b,c))
n=5: Should be impossible.
You can grind through the solutions to the binomial, trinomial and quartic equations. But when you try to do it with a quintic equation, at what point do you run up against a brick wall?
Here’s another TH-cam video explaining the general solution to a quintile equation: m.th-cam.com/video/SFXQ_920d4s/w-d-xo.html&pp=ygUOZ2Fsb2lzIHF1aW50aWM%3D You can use algebraic techniques to manipulate the equation into the form x^5 + x + a = 0, but after that, you need to use obscure things like hyper geometric functions to solve it.
Okay…can 3B1B pleeeeease do an in depth video on this? I am desperate for an intuitive explanation.
Fantastic podcast! Keep it up!
Well I found that tough. Thank goodness no adverts.
Grant is a real math scholar
Grant really is amazing
i watched many videos on this. i will never understand it until someone takes ax^5+bx^4+cx^3+dx^2+ex+f = 0 and using these terms shows why it cannot be solved algebraically.
This topic is very complex with a lot of ugly vocabulary. Just like this channel here, I also made a video trying to explain it in the shortest, most intuitive path that I can think of, although I’m no expert. Here’s the best idea behind it I've got:
1) A quintic formula, if it existed, would have to be able to have all 5 of its roots interchangeable, since Vieta’s equations show that the coefficients in such a formula would be symmetric. ALL FIVE roots would be interchangeable.
2) It’s possible that a quintic equation may have a root with another root nested inside of it. Perhaps more than one. This could cause huge complications with interchangeability. Each radical root would have conjugates, and swapping a root that’s nested in another root would really mess up the conjugates of the outer root if done carelessly.
3) To avoid this potential issue, the set of all roots of the polynomial would have to be arranged in such a way that no matter HOW you permute the entire set of all five of them, the subset of the roots nested inside of other roots would have to stay in its own set. Any set of roots like this fits the criteria of a normal subgroup. There may be normal subgroups that do not have this property, but if a set DOES have this property, it will be invariant under conjugation, just like a normal subgroup.
4) The nesting of radicals can be thought of as the nesting of subgroups. And every time there’s a set of roots nested inside of another set of roots, it would have to obey the property of staying inside of its own set, no matter how the full group is permuted.
5) If a group of roots is not a chain of normal subgroups ending in the trivial group, the property of roots staying in their own group is not guaranteed. S5, the group representing all possible arrangements of 5 elements (roots in this case) can possibly have subgroups that are not a chain of normal subgroups. The A5 group is one such subgroup. Since this is a possibility, the quintic is not solvable in radicals.
"You want to make the cubic depressed so that it's easier to work with." -Grant Sanderson
Is there not just for n》 4 just one example with one nonRadical root? I think this would simpel to falsify.
While this is a good video, I'd like to see Grant make one of his great polished videos on this topic. I still couldn't grok the end. BTW, I understand the Arnold proof pretty well.
This is great, just subbed!
Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s
buen contenido :)
It is a proof that for all his genius Grant is a mere mortal when he takes several minutes to notice that the sum of the two cubes in the cubic formula is constant no matter which cube root you take because you always have w^3 = (omega*w)^3 = omega^3*w^3 = 1*w^3 because omega is a third root of unity
The root x satisfies x = w + z. No 3rd power and so depends on which cube root you take.
@@TimothyNguyen At one point Grant wonders if the choice to make of the cube roots depends on the two conditions set (namely w^3+z^3 = -q, wz = -p/3) but the first one is always fulfilled no matter what choice you make because the different values of w and z differ by a scalar that is a third root of unity so the sum of the cubes is invariant and always equals -q. So only wz = -p/3 matters in the choice of the cube roots and you can bypass the issue of making a choice by setting z =-p/3w and since w has only three values x = w - p/3w only ever takes three values, the three roots of the cubic.
@@afuyeas9914 Yes we eventually arrived at that conclusion and it did take awhile. From my experience however, it is quite a different experience doing public math vs private math. In the former case, on my podcast, you have to juggle what you're saying and writing with what the other person is saying and writing (on the fly!), and for me, I additionally have to be alert of tech issues and keeping the podcast on track. Perhaps a more relatable situation would be having a teacher volunteer you to do math in front of the class vs doing math privately. Hopefully our fumblings at times are more instructive than painful to watch!
@@TimothyNguyen It was funny, if anything.
Which shared whiteboard S/W did you use in this video?
Google jamboard.
I am thinking about polynomials from a linear algebra perspective, and I was wondering what you think about looking at polynomials of degree 3 as being the abstract vector space of polynomials of degree 3 or less; which is to say that we imagine having a vector space having some basis. The next thought question would be if we look at each term of the vector space as being an abstract vector, than how many vectors we would need to fully describe the vector space of degree 3 or fewer. Taking Grant's idea of something plus something else, one may intuitively ask about how many sums or "linear combinations" where the "vectors" in our abstract vector space are the linear terms themselves. It would seem like then intuitively each solution would be essentially unique because if they were not, we may write 1 or more of the terms as a linear combination of the others. It would follow that if you did find a root as a solution but you only found 1 real number, than the other 2 solutions must be a different kind of number, given that 1 basis vector can not possibly describe the vector space we are trying to model degree 3 polynomials or less as basis vectors. If we would still need more basis vectors. I wanted to get your thoughts as to whether you believe this motivated the study of what ultimately lead to the fundamental theorem of algebra in reference to complex roots when it comes to finding the coordinates of where the basis vectors got mapped to. I am self studying math, and I am actually completely blind, so I have always been fascinated with applied and pure math. I always wondered if I new the appropriate isomorphism, if I can take anything I hear someone talk about geometrically and rephrase the perspective into something that is more accessible like linear algebra, number theory, combinatorics, or any other branch of mathematics that does not require the immediate need to draw a picture or look at something visual. My personal mission is to make math and science accessible to everyone even blind people. I am inspired by educational videos like yours
and Grant's, and keep up the great work!
This title confused me. f(x) = x^5 - 1 = 0 is a quintic and may be solved for five roots, which makes it solvable. Many complex quintic functions, expressed in the form f(x) = 0, may be factored or solved by various iterative methods. So should the title read something like "Unsolvability of Some Quintics?"
The (general) quintic is unsolvable.
Amazing
He is hot at voice and math ! Thank you ! Grant Sanderson.
The result of Galois theory is that there is no formula that provides solution(s) by inputting the coefficients. But there is the claim that proves the quintic is "unsolvable". That is a false claim. The result has not ruled out analysis of coefficient properties and properties of the function values. Find one solution and then there's only the quartic to be done.
the first man who introduces symbolism in mathematics was Alkhawarizmi see (al jabre wal mokabala)
interesting stuff
One prooves that An, n > 4 is a bad husband that does not love his wife. But how does somrone the idea to study automorphisms? And I would like somrone to find a simpler proof of the 121.
Grant, couldn’t the formulas for third and fourth order polynomials be derived by a linear transformation of the real and imaginary roots to transform the roots to be the third (or sixth) and the eighth roots, respectively, then the reconversion to the original scale. Obviously this approach does not work for quintic polynomials, in general.
UGs should maybe ask a simpler question first:
_Is there always a solution (real) to all polynomials of any degree?_
Well we are taught very early that we can answer in the quadratic case using the discriminant.
Then we realize that all Odd-order polynomials have at least one solution (they must cross the x-axis somewhere).
Then the remainder/factor theorem helps in some arbitrary cases.
Then after that we are in uncharted territory in deciding if there is a general answer to my question.
Obviously ALL Order-5 quintic polynomials can be factored with one linear term.
Maybe 3B1B (or someone here) would like to answer the question of quintic solvebility over the Quarternions or Octonian, Sedonian fields. Might be?
none of those are fields unfortunately
@@98danielray Can you briefly explain why these are not fields? Ignoring commutativity of products.
Is it because Octonians and so on are not associative?
Even if they are not "fields" we still need to know if f(X) =0 a polynomial has solution in these X domains.🥺
The Asian guy should talk less and write less and let the guy who is actually far more informed than him to talk and write. In short poorly done interview.
Insta-subscribed
Grant is wrong when he stresses that one "never" has to use the cubic formula - I know of at least two calculations in physics where it is actually necessary. (One in cosmology, for solving the Friedmann equations in a radiation-dominated universe, and one in quantum mechanics, when calculating the energies for an anharmonic oscillator perturbatively.)
He is saying " no one uses cardano's formula".
@@paperclips1306 And that claim is simply wrong. I gave two counterexamples.
if u had to end up using nonsense complicated group theory terms y even act like derivin things from fundamental concepts in the start..
I used to think he was the brown π 😮
Genius to write down the polynomial x^3-7x^2+11x-8=0 (not x^3-7x^2+14x-8=0) and claim the solution is 1, 2 and 4. People will watch the rest of the video to see if they pick up the mistake. No spoilers sry - the video is a good watch.
Tartaglia _does_ look pretty pissed off.
Only zoology people like animals and humans cleaning. Preservation conservation etc. symmetry is not there beyond 5.
Wtf bro
tl;dr: Never die in a duel (or a dual). 😕
I can't understand shit
He is not a mathematician
Check out NOT ALL WRONG for a really modern presentation which by passes the Galois development, entirely.