[HN Gopher] There is No Quintic Formula [video]
___________________________________________________________________
There is No Quintic Formula [video]
Author : DamnInteresting
Score : 35 points
Date : 2025-11-30 18:15 UTC (4 hours ago)
(HTM) web link (www.youtube.com)
(TXT) w3m dump (www.youtube.com)
| addaon wrote:
| Without video:
| https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
| sparky_z wrote:
| 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.
| throwaway150 wrote:
| > 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?
| cgriswald wrote:
| 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.
| addaon wrote:
| 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.
| addaon wrote:
| > 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.
| isotypic wrote:
| 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.
| pfdietz wrote:
| There is if one is allowed to use elliptic functions.
| susam wrote:
| 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.
| steppi wrote:
| 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.
| cyberax wrote:
| 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
| Paracompact wrote:
| 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.
| gblargg wrote:
| Thank you, much better; the visuals supplement the words rather
| than try to distract and wow me.
| logannyeMD wrote:
| 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
___________________________________________________________________
(page generated 2025-11-30 23:00 UTC)