But why is there no quintic formula? | Galois Theory

"The best way to learn a new topic is to teach it" - Grant Sanderson aka 3blue1brown

Galois theory is a fascinating topic and I hope you learned as much as I did while making this video.
Please leave me any feedback, questions, etc. in the comments.

===Chapters===
0:00 Introduction
0:24 Groups
1:18 Fields
3:03 The Connection
5:30 Solving a polynomial
9:42 Conclusion
10:46 Why is there no quintic formula
11:38 Outro

Ignore the typo at 0:14. It should be "Group Theory"

At around 3:25 The last equation should equal -2, not 2. Still it has the same result even if you swap the roots.

===Further material===

===Made with===

===Music===
Music by Vincent Rubinetti
Download the music on Bandcamp:
Stream the music on Spotify:

===Tags===
#galois #quintic #grouptheory #galoistheory #abelruffini

[mathkiwi]

Hi, I hope you liked my video!
Please leave your feedback, questions, etc. in the comments.
Further material is in the description

[altimpneo]

I've been watching several videos related to this topic (3Blue1Brown, Math Visualized, Aleph 0, etc.) and so far this video imo is the one that tries to explain the idea behind this concept in the cleanest way possible, the animations for this video are incredibly good and made me understand some concepts that I didn't get from other channels explaining the same topic
(found this video way more friendly in the given steps to build the group table, also the animations showing what do permutations do to the roots of a polynomial are so good) . Awesome video.

[diegogamba]

It is clear that you understand what you are talking about and that your goal is to produce videos that highlight the essence of a certain mathematical topic. If you intend to address only those who already know it, to show how elegantly these could be conceptually summarized, the style is good and the graphics are clean. However, if you are also addressing "the rest of us", who do not know a certain topic, or never understood it properly, to put them on the right track, I would suggest a slower pace. It can also be helpful to check how YouTube interprets your words. An example (4:54): "...color group of the polynomial of a q to the treble group the channel fields..." (= "Galois group.... trivial group... chain of fields... "). I would also suggest providing some information about your Channel. If you don't want to put any personal information in it, at least spell out your purposes. Clearly formulating what you want to do would be of benefit both for you and for your target audience. As you can see I'm interested in the kind of stuff you're doing and wish you all the best.

[runnow]

wait so there's no quintic formula because you can't always divide the permutations of a quintic ? Galois theory is the only thing I've seen like 3 videos on and am still lost lol

[theteleportercell]

This is so u underrated, your editing and/or manim skills are definitely at 3b1b's level, really deserved more subs man!

[Kasthurikannan]

At 3.25, the last equation should equal -2. Excellent video!

[Sharrrian]

This is honestly so well explained. I haven't been able to understand this for a few weeks until I saw your video, and now I'm crystal clear about this elegant proof. Subscribed and thank you!

[masonskiekonto]

Definitely one of the better videos on unsolvsbility of the quintic

[andrewkamoha]

Just a reminder if you are having trouble understanding all this:
Galois died at 20 years age.
And several months of his short life were imprisoned due political reason.

[Dark_Souls_]

This was an awesome video, thank you. I haven’t taken a class in abstract, only seen videos to cure my curiosity, and this was a great learning. Subscribed

[ooffoo]

This is the video.

Finally I understand....

Every other video I've watched was way too overexplained, this was perfect.

[Kasthurikannan]

Relating this to the fundamental theorem of symmetric polynomials that relates coefficients to the roots will be fantastic. Why should roots obey the permutations in the first place for solvability? Because if they don't we will have the original polynomial altered.

[CardiganBear]

Thanks for this. I have been trying to understand this topic for some time and have watched a number of videos about it. This one however, I find offers the best hope of gaining a grasp of the essential steps in the argument in a clear way without going into a lot of unnecessary detail.

[Caturiya]

DIDACTICALLY BEST VIDEO OF GALOIS THEORY. Nice your circle graphics !!

[romanvolotov]

good 3b1b style with a speedier narrative, i'm a fan

[Revominded]

I have elementary background of abstract algebra from years ago. You made galois theory a bit easier to digest, I still can’t grasp it but I can at least understand a bit of the premise of why Quintics and above don’t have formulas

[Zarunias]

Does all of that mean that there is still hope that one day we find general solutions that includes trigonometry for polynomials of higher degrees? Somewhat like the "casus irreducibilis" for 3rd degree polynomials?

[matthewkendall]

I once put a general form 10th order polynomial into Cayley in 1983 to see how it would handle it - it solved in under a second - but the formulea was 10 pages of print out! Amazing what a Group Theory program designed for infinite precision maths and to handle groups of up to 10 ^ 50 elements could do even back then!

[holyshit]

One remark - unsolvable by radicals
Once we allow functions like hypergeometric functions or stuff like this
polynomials will be solvable

[hyperduality]

Subgroups (discrete, quantum) are dual to subfields (continuous, classical) -- the Galois correspondence.
The size of a Galois group measures the amount of symmetry of the roots of a polynomial -- symmetry breaking!
Randomness (entropy, lack of symmetry) is dual to order (syntropy, symmetry) -- Galois groups.
Symmetric polynomials have large Galois groups.
Symmetry wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- quantum duality.
"Always two there are" -- Yoda.
Symmetry is dual to conservation (invariance) -- the duality of Noether's theorem.
Patterns, symmetry = predictability, certainty or syntropy!