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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.

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

There is if one is allowed to use elliptic functions.

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.

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.

Thank you, much better; the visuals supplement the words rather than try to distract and wow me.