The Insolvability of the Quintic

This video is an introduction to Galois Theory, which spells out a beautiful connection between fields and their Galois Groups. Using this, we'll prove that the quintic has no general formula in radicals.

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SOURCES and REFERENCES for further reading!

As with any quick introduction, there are details that I gloss over for the sake of brevity. If you’d like to learn these details more rigorously, I've listed a few resources down below.

(a) Group Theory

This playlist by Professor Benedict Gross is a beauty. It goes through the entire group theory syllabus from the ground up, and Professor Gross is a masterful lecturer.

(b) Galois Theory

“Galois Theory” by David Cox is a skinny little book that goes through the main theorems of Galois Theory. The first few chapters give historical background, and the remaining chapters lay out the key theorems and applications.

“Galois Theory for Beginners” by Jorg Bewersdorff explains the insolvability of the quintic in intuitive terms. It doesn’t assume any prior background knowledge, and all chapters but the last can be understood without group theory. The last chapter formulates the theorem using the language of groups and field extensions, but it explains all the definitions as it goes along.

This playlist on Galois Theory by Professor Richard Borcherds is a gem. It explains Galois Theory from the ground up, rigorously, in almost complete generality.

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SOME NOTES ABOUT THE VIDEO CONTENT:

Galois Theory is normally introduced at the end of a course in abstract algebra, and for good reason. There’s a lot of technical machinery involved, and I’ve deliberately omitted certain parts that I felt were not immediately relevant towards the insolvability of the quintic. If you’re interested in seeing how the ideas in this video differ from the standard treatment, read on.

1) (The Galois Group.) In this video, we define the Galois Group of a *polynomial*. In the modern treatment, however, we normally talk about the Galois Group of a *field extension* (not a polynomial), and we define it as the set of all automorphisms of the top field that fix the bottom field pointwise. When I refer to the Galois Group of a polynomial, I am referring to the Galois Group of its *splitting field*, viewed as a field extension of the rational numbers. But obviously, that’s quite a mouthful. That’s why I took the route I did; I felt that introducing all this machinery – automorphisms, splitting fields, etc. – would have obscured the main point of the video.

2) (Normal Subgroups.) We observed in the final example that the subgroup partitions the group table into squares, and many of the squares had the same elements, just in a different order. A subgroup that splits the group table into squares so that any two squares are either equal or mutually disjoint is called a “normal subgroup”.

This is a non-standard definition of a normal subgroup – although, it is equivalent to the standard definition. I felt that this definition of a normal subgroup was a lot more intuitive than the standard definition (which, for the record, I still find quite mysterious, even after having taken a course in group theory!)

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MUSIC CREDITS:

Song: Thinking Ahead

SOFTWARE USED:
Adobe Premiere Pro for Editing

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Intro: (0:00)
Field Extensions: (0:48)
Galois Groups: (3:22)
The Insolvability of the Quintic: (8:20)

[ultravidz]

One of the few aspects of YouTube that improves with time is the growing audience for content like this. It has always been a minuscule percentage of users, but with the growth of the user base there’s enough viewership to reward and motivate content creators like this. I’ll never take this for granted.

[muesk]

I agree with the sentiment that perhaps too large of a jump was taken for the relationship between the tiling and the lack of formula. I followed everything before, and that step came seemingly out of nowhere. I couldn't believe the video was already over, made me feel like something was missing.

[Tsskyx]

Nice explanation, although I'm slightly confused since you didn't explain everything. Where do the subgroup chains come from and what do they represent? I think you should show an example with concrete polynomials. And also, although this is probably a more advanced topic, I'd like to know why exactly do non-prime numbers mean that it's not solvable, since it seems that the entire proof hinges upon this single assumption.

[sacralv]

Bro, I hadn't checked your channel in a while and I notice there are two videos missing:
-How to learn pure mathematics
-Math isn't ready to solve this problem
What happened to those videos:(? I really loved them.

[GFJDean]

8:00 "In general, if a polynomial is solvable by radicals the number of tiles is a prime number." But why is this the case? This is kind of a conceptual gap in the video that left me dissatisfied.

Great video otherwise!

[k_meleon]

Dude. I was reading the section about galois theory literally today in Modern Abstract Algebra and I didn't get it at all lol... Anyway, thank you so much for these videos, you truly are a hidden gem of youtube

[renatodiascosta]

When this channel explains something I already know, somehow it makes me know it better

[thomasidzikowski]

I wish they had your video during my Abstract Algebra days at Uni. The "roadmap" of a mathematics course at this level is generally obscured by the cramming of fundamental concepts into our heads, not how those concepts have built the map.

Thank you for filling in a fundamental part of the map.

[antonseoane]

👏👏👏 just made sense of a full course on Galois Theory full of formalizations of automorphisms and company and empty of intuition. Great video.

[Icenri]

I've seen this topic a bunch of times but never going deep enough and without this level of clarity. Massive thanks.

[gonzalochristobal]

Okay man I have to say that I've watched and read a fair amount of introductory content to Groups, Abstract Algebra and Field Extensions and this is by far the most comprehensive one. It is amazing how clearly you present the ideas, an statement of how organized you have the topic in your own mind, truly amazing! Thanks!

[adsoyad]

there's nothing like the feeling of discovering a new maths channel. great content you're severely under-appreciated

[kerbalspaceinstitute]

Although there may be a detail or two that you did not explain, still you break the discussion down to the crux of the matter and thus make the discussion accessible to the uninitiated. That is inspiring! Now I really want to learn the details.

Thank you 😊

[nicolararesfranco]

Great job!! I always wondered how that worked out. As I specialized in functional analysis and probability theory, I never got the chance to dive deeper into this stuff. Thank you!

[rajdeepghosh]

Great job Aleph! Been waiting for this for a while

[Tomyb]

The "why" you never get a prime tiling beyond 4 seems just as mysterious as to why there's no general formula for beyond quintics.

[mr.bulldops]

I've not taken any analytic algebra courses and still able to follow along with the basics thanks to your great visuals and explanations.

[iankr]

Many thanks. I did a course on Galois Theory in the third year of my maths degree, 40 years ago. I can't recall much of it, but this video has brought bits of it back to me. This has whetted my appetite to read up on it.

[dmr]

Love your videos! A recent problem set of mine involved proving that A_n is simple for n>4 and this is making me excited to learn Galois theory this spring

[goofmuffin]

i see a lot of people in the comments here who find this explanation unsatisfying, and this is certainly a good instinct to have! the reason is however because to actually understand this result you need a courseful of context, and a 10 minute youtube video can only do so much. i don't think this video was ever supposed to be a comprehensive explanation, but just an introduction to the topic c:

btw the way the music loops is killing me