There is No Quintic Formula [video]
youtube.com
youtube.com
If he defines integers as "natural numbers excluding zero," that seems goofy and nonstandard but also interesting. Is that a Russian-specific convention?
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.
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?
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.
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.
There is a much better video by a real human: https://www.youtube.com/watch?v=BSHv9Elk1MU
Personally, I think 2swap is the best math education channel to come by since 3blue1brown.
...Maybe there should be more music in math class?
We get more of what we tolerate, so let's not tolerate that.