@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!
@Mutual_Information
Evariste Galois was so impressive.. especially considering the fact that he invented all this before he died at 20 years old
@jburfoot
This is the best explanation of why the Galois correspondence implies quintic insolvability that I've seen.
@johnchessant3012
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!
@DaveJ6515
Brilliant exposition. And, as it happens when it's about real mathematics, only a few numbers were mentioned: 1,2,3,4,5.
@mathemaniac
[The pinned comment got removed by YouTube, again...] This is a very ambitious video, and it took me a lot of time and effort - please like, subscribe, comment, and share this video! If you can, please support the channel on Patreon: https://www.patreon.com/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.
@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!
@Adityarm.08
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 :)
@Ben777World
You made this about as clear as possible without a full course. I have definitely gained insight into this difficult area.
@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!
@sideral
Awesome . Understanding this deeply is one of the things i want to do before I die ☺️. Nice video!
@charleschidsey2831
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.
@0ddSavant
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!
@robokaos69
Abstract math is difficult for me. I appreciate high quality videos such as yours to help mitigate that struggle :)
@MatildaHinanawi
This is SO hard but I've been so curious about it for ages, props on the video!
@LucasDimoveo
I hope this channel gets more views. The editing and audio quality are fantastic
@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
@456MrPeople
I've been looking for a video like this for a long time. You're the next 3blue1brown!
@marcoskano4437
As of right now I am a week before my final exam of the second class on this topic and yet this video have helped me more than a lot of the notes and lecture at school. Thanks!
@mathemaniac