Vladimir Arnold famously taught a proof of the insolubility of the Quintic to Moscow Highschool students in the 1960s using a concrete, low-prerequisite approach. His lectures were turned into a book Abel’s Theorem
in Problems and Solutions by V.B. Alekseev which is available online here: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/abel.pdf. He doesn't consider Galois theory in full generality, but instead gives a more concrete topological/geometric treatment. For anyone who wants to get a good grip on the insolubility of the quintic, but feels overwhelmed by the abstraction of modern algebra, I think this would be a good place to start.
Looks like a nice book, but what's up with his assertion on page 148 (164 of the .pdf) that the integers don't form a group under addition?
If he defines integers as "natural numbers excluding zero," that seems goofy and nonstandard but also interesting. Is that a Russian-specific convention?
It seems like a typo where "integers" is used when the intention was to write "natural numbers". That is the solution to exercise 194 part a) which asked if the set of natural numbers is a field.
Whether 0 is a natural number is still fairly ambiguous; I remember being taught (1990s UK) to be specific about which definition was being used, or to prefer another name such as 'positive integers' or 'non-negative integers'
I learnt this subject from the book Galois Theory, 5th ed. by Ian Stewart. Quoting from page 177:
Theorem 15.10. The polynomial t⁵ - 6t + 3 over ℚ is not soluble by radicals.
As you can see, this theorem occurs in Chapter 15. So it takes fourteen chapters before we reach here. It takes a fair amount of groundwork to reach the point where the insolubility of a specific quintic feels natural rather than mysterious.
To achieve this result, the book takes us through a fascinating journey involving field extensions, field homomorphisms, impossibility proofs for ruler and compass constructions, the Galois correspondence, etc. For me, the impossibility proofs were the most interesting sections of the book. Before reading the book, I had no idea how one could even formalise questions about what is achievable with a ruler and compass, let alone prove impossibility. Chapter 7 explains this beautifully and the algebraic framework that makes those proofs possible is very elegant.
By the time we reach the section about the insoluble quintic, two key results have been established:
Corollary 14.8. The symmetric group S_n is not soluble for n ≥ 5.
Theorem 15.8. Let f be a polynomial over a subfield K of ℂ. If f is soluble by radicals, then the Galois group of f over K is soluble.
The final step is then quite neat. We show that the Galois group of f = t⁵ - 6t + 3 over ℚ is S₅. Corollary 14.8 tells us S₅ is not soluble. By the contrapositive of Theorem 15.8, f is not soluble by radicals.
Obviously whatever I've written here compresses a huge volume of work into a short comment, so it cannot capture how fascinating this subject is and how all the pieces fit together. But I'll say that the book is absolutely wonderful and I would highly recommend it to anyone interested in the subject. The table of contents is available here if you want to take a look: https://books.google.co.uk/books?id=OjZ9EAAAQBAJ&pg=PT4
Two small warnings: The book contains a fair number of errors which can be confusing at times, though there are plenty of errata and clarifications available online. And unless you already have sufficient background in field homomorphisms and field extensions, it can take several months of your life before you reach the proof of the insoluble quintic.
2swap makes some fantastic videos, I'd recommend giving them a follow on YT if you enjoy math visualizations. They also seem to spend quite a bit of time on the audio for each upload
This video is truly remarkable. I'm so grateful to artists like 2step for sharing this kind of work on YouTube. It reignites a passion for math that many of us might have forgotten, especially those of us who have been away from formal math education for a while.
This is some high quality content. Love the visual animations to go along with the mathematical ideas. Did a great job helping to tie the algebra to geometric intuition, but I think the importance of commutators could have gotten a little bit more exposition.
This is one of those situations where the video is just an insane value-add above and beyond the Wikipedia article that this sort of response is baffling to me. The well thought out presentation and progression of the concepts. Just enough context to keep the non math grad students following along without wasting time or talking down to the audience.The incredible visualizations that are both beautiful and insightful. Someone spent months of their life making this video as good as it could be, and it shows.
> This is one of those situations where the video is just an insane value-add above and beyond the Wikipedia article that this sort of response is baffling to me. The well thought out presentation and progression of the concepts.
This is good to know, for this video. Unfortunately, HN doesn't have a way to indicate this other than linking to a YouTube video; and in my experience very few YouTube videos are a superior way to absorb information than reading. To find that out, I'd have to either watch the video (negative expected value), or wait for a comment from someone like you -- and now that the latter has happened, perhaps I'll actually try to watch it. In the meantime, I do think there's value in providing information without a (sometimes literal) song and dance around it for those interested in learning over entertainment, on average.
All you have done is contribute a wikipedia article which is the second google result if you search the title of the video. Another user made a comment referencing a textbook they used to learn this material as well as some extended comments of their own - this actually provides information unlike a bare wikipedia link presented with a dismissive attitude.
I'm struggling to understand the negative tone in your reply to the parent comment. They simply offered an additional resource on the topic. Rather than welcoming it, you seem to have taken issue with it. One of the strengths of HN threads is that people often contribute further material that others may find helpful.
The video is useful but so is the Wiki article. Some readers will prefer the video, some the article, and some both. Why object to someone sharing another link?
In fairness to the GP, the OP has now admitted that they made the post without having watched the video and that they did so out of prejudice against YouTube videos. GP wasn’t objecting to the additional resource but the implication via “Without video:” that the video itself is less valuable.
As the OP, I agree with everything you said, but I suggest an alternate characterization: Some subset of people, including me, prefer written communication to video (regardless of whether the video is on YouTube or elsewhere). Since my favorite HN threads delve into a topic, rather than into the details of a particular presentation of a topic, and since on seeing this topic raised I hopped over to Wikipedia to refresh my memory on this topic, I thought I would provide a breadcrumb for others of similar mindset to help jumpstart the topical discussion. Which, clearly, I was not quite successful in doing -- so, lesson learned.
The current top comment by u/steppi is a stellar example of how to offer an additional resource on the topic in a way that adds value to the discussion. This was not that.
I (correctly) interpreted the terse, dismissive tone of their comment, which implied that the video added no value beyond what was found in the Wikipedia page. Other comments here confirm that I'm not the only one who read it that way. The clear subtext was "don't waste your time with that slop, just read this." I was certain that, if that was their takeaway, then they hadn't even bothered to watch the video (which turned out to be correct). But I only knew that because I had already watched the video the previous day and was deeply impressed by it.
At the time I replied, it was the only comment here, and was therefore setting the tone of the discussion. I didn't want all the people who only follow the link after checking the comments to assume that the video was just a lazy Wikipedia summary, upvote the comment for "saving them time", and then move on. My primary goal was to actually describe the video and encourage anyone out there who likes this sort of thing to give it a shot. In order to do that effectively, I felt I also had to push back on the impression left by the comment I replied to.
How is this the same thing but without video? This doesn't even mention the unique visualization used in the video, which most people familiar with this topic have never seen before.
It's understandable to hold different perspectives on the video. However, simply dismissing it as "AI" without any supporting evidence or referring to the audio as "noise" is unproductive and doesn't foster a constructive discussion.
There might as well have been. What an annoying video to watch (and listen to). Did they play music in the math classroom when this guy went to school? No? Well, what's it doing here?
What an annoying comment to read about such an incredible video. Both me and my son enjoyed it a lot. It's an educational art piece with beautiful and insightful visualizations.
Seriously, what's incredible about it? I know it's not AI, and I admittedly did not watch the whole thing, but from what I saw, there is absolutely no element of that video that couldn't be auto-generated at this point.
The sound design complements it extremely well. It's a lot like that classic sphere eversion film Outside In produced by The Geometry Center, which uses different sound effects to build up the steps in the explanation, subtly helping the audience not lose track of what's going on.
...Maybe there should be more music in math class?
It will be even more annoying if more and more otherwise-good educational video producers start to add gratuitous, distracting music to everything they release.
We get more of what we tolerate, so let's not tolerate that.
You really picked a wrong target for your ridiculous attacks. 2swap is a mathematical artist, not Khan Academy. There's plenty of boring lectures on youtube already on every topic.
Vladimir Arnold famously taught a proof of the insolubility of the Quintic to Moscow Highschool students in the 1960s using a concrete, low-prerequisite approach. His lectures were turned into a book Abel’s Theorem in Problems and Solutions by V.B. Alekseev which is available online here: https://webhomes.maths.ed.ac.uk/~v1ranick/papers/abel.pdf. He doesn't consider Galois theory in full generality, but instead gives a more concrete topological/geometric treatment. For anyone who wants to get a good grip on the insolubility of the quintic, but feels overwhelmed by the abstraction of modern algebra, I think this would be a good place to start.
This book is also an accessible introduction to group theory, I managed to work through the first half of the book when I was 15 y.o.
Looks like a nice book, but what's up with his assertion on page 148 (164 of the .pdf) that the integers don't form a group under addition?
If he defines integers as "natural numbers excluding zero," that seems goofy and nonstandard but also interesting. Is that a Russian-specific convention?
It seems like a typo where "integers" is used when the intention was to write "natural numbers". That is the solution to exercise 194 part a) which asked if the set of natural numbers is a field.
Whether 0 is a natural number is still fairly ambiguous; I remember being taught (1990s UK) to be specific about which definition was being used, or to prefer another name such as 'positive integers' or 'non-negative integers'
notation has unified dramatically since the 1960s.
Yes, I think there were copies circulated (not xerox but blueprints)
I learnt this subject from the book Galois Theory, 5th ed. by Ian Stewart. Quoting from page 177:
Theorem 15.10. The polynomial t⁵ - 6t + 3 over ℚ is not soluble by radicals.
As you can see, this theorem occurs in Chapter 15. So it takes fourteen chapters before we reach here. It takes a fair amount of groundwork to reach the point where the insolubility of a specific quintic feels natural rather than mysterious.
To achieve this result, the book takes us through a fascinating journey involving field extensions, field homomorphisms, impossibility proofs for ruler and compass constructions, the Galois correspondence, etc. For me, the impossibility proofs were the most interesting sections of the book. Before reading the book, I had no idea how one could even formalise questions about what is achievable with a ruler and compass, let alone prove impossibility. Chapter 7 explains this beautifully and the algebraic framework that makes those proofs possible is very elegant.
By the time we reach the section about the insoluble quintic, two key results have been established:
Corollary 14.8. The symmetric group S_n is not soluble for n ≥ 5.
Theorem 15.8. Let f be a polynomial over a subfield K of ℂ. If f is soluble by radicals, then the Galois group of f over K is soluble.
The final step is then quite neat. We show that the Galois group of f = t⁵ - 6t + 3 over ℚ is S₅. Corollary 14.8 tells us S₅ is not soluble. By the contrapositive of Theorem 15.8, f is not soluble by radicals.
Obviously whatever I've written here compresses a huge volume of work into a short comment, so it cannot capture how fascinating this subject is and how all the pieces fit together. But I'll say that the book is absolutely wonderful and I would highly recommend it to anyone interested in the subject. The table of contents is available here if you want to take a look: https://books.google.co.uk/books?id=OjZ9EAAAQBAJ&pg=PT4
Two small warnings: The book contains a fair number of errors which can be confusing at times, though there are plenty of errata and clarifications available online. And unless you already have sufficient background in field homomorphisms and field extensions, it can take several months of your life before you reach the proof of the insoluble quintic.
2swap makes some fantastic videos, I'd recommend giving them a follow on YT if you enjoy math visualizations. They also seem to spend quite a bit of time on the audio for each upload
The 2swap lambda calculus video (https://www.youtube.com/watch?v=RcVA8Nj6HEo) is a masterpiece!
This video is truly remarkable. I'm so grateful to artists like 2step for sharing this kind of work on YouTube. It reignites a passion for math that many of us might have forgotten, especially those of us who have been away from formal math education for a while.
*2swap
There is if one is allowed to use elliptic functions.
TIL https://math.stackexchange.com/questions/540964/how-to-solve...
This is some high quality content. Love the visual animations to go along with the mathematical ideas. Did a great job helping to tie the algebra to geometric intuition, but I think the importance of commutators could have gotten a little bit more exposition.
Without video: https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
This is one of those situations where the video is just an insane value-add above and beyond the Wikipedia article that this sort of response is baffling to me. The well thought out presentation and progression of the concepts. Just enough context to keep the non math grad students following along without wasting time or talking down to the audience.The incredible visualizations that are both beautiful and insightful. Someone spent months of their life making this video as good as it could be, and it shows.
> This is one of those situations where the video is just an insane value-add above and beyond the Wikipedia article that this sort of response is baffling to me. The well thought out presentation and progression of the concepts.
This is good to know, for this video. Unfortunately, HN doesn't have a way to indicate this other than linking to a YouTube video; and in my experience very few YouTube videos are a superior way to absorb information than reading. To find that out, I'd have to either watch the video (negative expected value), or wait for a comment from someone like you -- and now that the latter has happened, perhaps I'll actually try to watch it. In the meantime, I do think there's value in providing information without a (sometimes literal) song and dance around it for those interested in learning over entertainment, on average.
All you have done is contribute a wikipedia article which is the second google result if you search the title of the video. Another user made a comment referencing a textbook they used to learn this material as well as some extended comments of their own - this actually provides information unlike a bare wikipedia link presented with a dismissive attitude.
> this sort of response is baffling to me
I'm struggling to understand the negative tone in your reply to the parent comment. They simply offered an additional resource on the topic. Rather than welcoming it, you seem to have taken issue with it. One of the strengths of HN threads is that people often contribute further material that others may find helpful.
The video is useful but so is the Wiki article. Some readers will prefer the video, some the article, and some both. Why object to someone sharing another link?
In fairness to the GP, the OP has now admitted that they made the post without having watched the video and that they did so out of prejudice against YouTube videos. GP wasn’t objecting to the additional resource but the implication via “Without video:” that the video itself is less valuable.
As the OP, I agree with everything you said, but I suggest an alternate characterization: Some subset of people, including me, prefer written communication to video (regardless of whether the video is on YouTube or elsewhere). Since my favorite HN threads delve into a topic, rather than into the details of a particular presentation of a topic, and since on seeing this topic raised I hopped over to Wikipedia to refresh my memory on this topic, I thought I would provide a breadcrumb for others of similar mindset to help jumpstart the topical discussion. Which, clearly, I was not quite successful in doing -- so, lesson learned.
The current top comment by u/steppi is a stellar example of how to offer an additional resource on the topic in a way that adds value to the discussion. This was not that.
I (correctly) interpreted the terse, dismissive tone of their comment, which implied that the video added no value beyond what was found in the Wikipedia page. Other comments here confirm that I'm not the only one who read it that way. The clear subtext was "don't waste your time with that slop, just read this." I was certain that, if that was their takeaway, then they hadn't even bothered to watch the video (which turned out to be correct). But I only knew that because I had already watched the video the previous day and was deeply impressed by it.
At the time I replied, it was the only comment here, and was therefore setting the tone of the discussion. I didn't want all the people who only follow the link after checking the comments to assume that the video was just a lazy Wikipedia summary, upvote the comment for "saving them time", and then move on. My primary goal was to actually describe the video and encourage anyone out there who likes this sort of thing to give it a shot. In order to do that effectively, I felt I also had to push back on the impression left by the comment I replied to.
How is this the same thing but without video? This doesn't even mention the unique visualization used in the video, which most people familiar with this topic have never seen before.
2swap is a legend! I highly recommend taking a look at his other videos as well.
Ugh. This video is an AI hell with distractions and an awful background noise (sorry, it's not music).
There is a much better video by a real human: https://www.youtube.com/watch?v=BSHv9Elk1MU
There is no AI here. You can even audit his mathematics and video rendering code here: https://github.com/2swap
Personally, I think 2swap is the best math education channel to come by since 3blue1brown.
Sorry about mis-AI-ing them. But still, the video was unwatchable for me.
It's understandable to hold different perspectives on the video. However, simply dismissing it as "AI" without any supporting evidence or referring to the audio as "noise" is unproductive and doesn't foster a constructive discussion.
There might as well have been. What an annoying video to watch (and listen to). Did they play music in the math classroom when this guy went to school? No? Well, what's it doing here?
What an annoying comment to read about such an incredible video. Both me and my son enjoyed it a lot. It's an educational art piece with beautiful and insightful visualizations.
Seriously, what's incredible about it? I know it's not AI, and I admittedly did not watch the whole thing, but from what I saw, there is absolutely no element of that video that couldn't be auto-generated at this point.
The sound design complements it extremely well. It's a lot like that classic sphere eversion film Outside In produced by The Geometry Center, which uses different sound effects to build up the steps in the explanation, subtly helping the audience not lose track of what's going on.
...Maybe there should be more music in math class?
Except if you don't like that music, then it just ends up being distracting.
What an annoying comment to read.
It will be even more annoying if more and more otherwise-good educational video producers start to add gratuitous, distracting music to everything they release.
We get more of what we tolerate, so let's not tolerate that.
You really picked a wrong target for your ridiculous attacks. 2swap is a mathematical artist, not Khan Academy. There's plenty of boring lectures on youtube already on every topic.
Stand-up maths is also an artist. Just as many other math channels.
Thank you, much better; the visuals supplement the words rather than try to distract and wow me.