Prelude to Galois Theory: Exploring Symmetric Polynomials

A short lecture explaining the fundamental theorem on symmetric polynomials and its relationship to Galois theory.
Reference book: Harold M Edwards - Galois Theory
Chapters:
00:00 Introduction
05:00 Definition 1 - Polynomial
07:36 Definition 2 - Symmetric Polynomial
08:28 Definition 3 - Elementary Symmetric Polynomials
10:01 Power Sum Theorem - Preamble
12:08 Power Sum Theorem - Proof
17:39 Fundamental Theorem on Symmetric Polynomials - Preamble
20:04 Fundamental Theorem on Symmetric Polynomials - Proof
28:00 Outlook to Galois Theory
30:32 Outro

[@videos-de-fisica]

Can't believe I just found maths' Bob Ross, what a great day

[@vNCAwizard]

The other end of it is that many, many people will not get the Bob Ross reference.

[@hydose1]

Your comments about why you did this video is the thing every teacher should understand. Without telling why those abstractions came about mathematics is being taught. A context would delineate the whole subject much better. An amazing video. You should do more of these

[@martintrifonov]

A very kind comment, thank you :)

[@roberthorne9597]

​@@martintrifonov Great work though! Deserved Do Godelllllll lol

[@TomSkinner]

I agree with this wholeheartedly. It's something that has always bugged me about the way math is usually taught.

[@clickaccept]

He writes on the board with hands in pockets. Magic.

[@Ricocossa1]

I love your editing, perfect handwriting, perfect pace. I don't know anything about Galois theory so this is gold. You've earned my subscription, and I'm looking forward to the follow-up! 😊

[@martintrifonov]

Thanks, I’m glad you liked it :)

[@nickush7512]

Dude, I am not a math type, and I am only a few minutes in, and already I am extremely imptessed with every aspect of you presentation. Congratulations from an educated leyman on your project, which is first rate as far as I am concerned. EDIT. Wow, I got through to 27 minutes before brain called Time-Out !! Superb work Dude.

[@martintrifonov]

Thank you, that’s very nice - I’m glad you enjoyed it :)

[@Achrononmaster]

What a quirky little thing you are. I'm happy to be making the first comment, and hoping you do a few more of these MMM's.Theoretical physicist here, semi-retired, doing things for the fun of it, but never really have time or energy to read dense mathematics tomes, so I really rely on the odd rare engaging bits of mathematics on youtube to keep in the game a bit. So thanks.

[@martintrifonov]

Thanks, I’m glad you enjoyed it. I appreciate the encouragement!

[@tempiadem586]

Amazing video! I learned briefly about Galois Theory in a history of math class, and I couldn’t understand the motivation for so many concepts that were introduced. This video was engaging the whole way through and I have so much more appreciation for symmetric polynomials! Really hoping for a follow up video!

[@joevogl3431]

Superb presentation in every aspect. Would have been nice 50 years ago when I first encountered Galois theory.

[@martintrifonov]

What a nice comment, thank you. Glad you enjoyed it!

[@kyle1977xy]

I love how you highlight the essence of galois theory and hence demystify it. Best video so far on symmetric polynomial . Incredible work !!! I'm eagerly looking forward to what comes next in group theory

[@jlas0324]

Impossible not to be humbled how a 20 years old guy from the early 1800s could come out with such a deep and abstract insight into algebra. Excellent job presenting the fundamentals of that insight, Martin, so concise and clear. Congrats!!

[@martintrifonov]

Humbling indeed. Thank you for the kind words!

[@arezaajouneghani3082]

This presentation, undoubtedly, stands as the quintessence of introductory discourse on this subject matter. The presenter undoubtedly possesses a prodigious intellect akin to that of Galois. Remarkably exceptional!

[@MrSamwise25]

Wonderful video! I've already taken a galois theory class, but I had the same frustration you described at the end: I didn't understand where all these definitions and proofs were coming from. This video reignited my intrigue for it. I especially liked your proofs. You gave just enough detail to give a full understanding without being slowed down, and you placed emphasis on the magical moments. It was really enjoyable!

[@martintrifonov]

I'm glad you enjoyed the proofs, it means a lot to hear that - thank you :)

[@sydneythesurfboards5903]

Hello Martin! Algebraist here. I would like to stretch out a hand and say that you did this presentation on symmetrical polynomials very wonderfully. Very clear. Very insightful. I’m looking forward to more of your videos! I might even have a thing or two to learn from you… 😉 Greetings from Sweden! 🇸🇪

[@martintrifonov]

Hey! Thank you for your kind words, I'm just starting out, so it means a lot. Greetings from Berlin!

[@glumbortango7182]

Sincerely thank you for trying to make this topic more approachable, I know from experience that it's not easy, and I cherish every resource I have that can show some insight on it. Edit: 31:00 This resonates with me a lot, I have asked myself many times with Galois Theory why anything shown was thought of, or how any of it follows from the axioms. Most of what I've seen was either too vague to actually show the specifics, or too technical to clearly explain the underlying material, so you're doing a great service by laying this out fully.

[@martintrifonov]

Thank you, that’s very kind - I’m glad you enjoyed it :)

[@mohammadareeb1289]

Awesome video!! Would love to see a follow up

[@robharwood3538]

Excellent use of exposition (telling the story of it) to illuminate a frustratingly slippery path towards Galois Theory. At least now we can see where we are stepping, and place our feet more firmly on the ground before us! Thank you! PS: Great idea using a 'green board' to present your formulas! A nice compromise between the slow-but-friendly blackboard/whiteboard, and the fast-but-impersonal use of math-formula animations! Very innovative!

[@thomasthorbjrnsen5026]

This was such a wonderful watch. I can't wait to see what else you are planning to make.

[@vladthemagnificent9052]

I am already familiar with the material but I still watched pretty much all the way through. It is just satisfying to watch

[@martintrifonov]

Glad you enjoyed it! Thanks, that really means a lot :)

[@roberthorne9597]

It was weirdly satisfying returning to the subject

[@grinchsimulated9946]

As a highschooler with an interest in cryptology, Galois Theory has been a puzzle I've been poking at for a while. Though we have not quite gotten to Galois Fields yet, this is definitely the clearest explanation I've seen of such concepts. Really hoping this will be a series

[@zacharysmith4508]

If you haven't done so already you'll need to familiarize your self with abstract algebra and a decent understanding of proofs beforehand.

[@imrematajz1624]

Hey, this is a brilliant introduction, easily missed or overlooked, but more and more enlightening the more you listen to it. The fog is lifting and the relationship between the Galois theory and the Representation / Group theory is becoming apparent. I think I am going to revisit this intro a couple of times more. Thanks!

[@martintrifonov]

Thanks, that’s really encouraging! Glad you found it useful!

[@willyballmann3589]

Great job. Well done. Better in terms of didactic value than many university lecturers!

[@martintrifonov]

Thank you, I'm glad you enjoyed it! :)

[@pthisthis]

Fantastic video. Very clearly presented and motivated. Thanks.

[@MrNoisypeople]

I need more of those! Really helpful

[@Arturo1404_]

Learning math is way easier with this kind of content, thank you!!

[@Nibor999]

Love your work! Thoroughly enjoyable.

[@0xero7]

That was brilliant! Hoping for more Chapters!

[@martintrifonov]

Thanks!

[@mlliarm]

Loved your intro. Decided to stay. Thank you for this.

[@martintrifonov]

Thats very kind, thank you! :)

[@eduardoeller183]

Excited for your next videos!

[@briansmith7458]

Will try again later. Great presentation.

[@In2Preppy]

Very clean and straight forward explanations of a very advanced and intricate subject. Well executed. And i like the format of the video.

[@martintrifonov]

Thank you, I appreciate the kind words! :)

[@punditgi]

Brilliant video! ❤🎉😊

[@niconeuman]

It took me around 3 nights to watch the video (as I watch KZbin usually before sleep), and I really enjoyed it. I should watch it again in the afternoon with a piece of paper! It was very interesting and excellently animated and presented. Congratulations! I would enjoy in the future a video centered in some applications, if you have time and interest in doing so! Thank you very much!

[@martintrifonov]

I'm glad you enjoyed it, your kind words mean a lot! Thank you!

[@panchananpramanik8276]

Excellent.

[@rchas1023]

Thank you for a lucid presentation of how to develop theories.

[@martintrifonov]

Glad it was helpful!

[@metiscus]

This is amazing! Please continue and make more videos!

[@martintrifonov]

Thank you for your kind words :) I will!

[@GelidGanef]

This is a really beautifully edited and crafted little video ❤️ I hit the bell 🔔

[@martintrifonov]

Thank you, that’s really kind :)

[@richardpalme5b]

Thank you, I enjoyed watching this video very much. This video convinced me to spend more time with pure math in the future, even though I am employed as a computer scientist and hence need to spend most of my learning time with IT topics :)

[@martintrifonov]

Thank you, that’s very kind - I’m glad you liked it :)

[@goncalofreitas2094]

I love your approach to math youtube videos, keep it coming 💪

[@martintrifonov]

Thank you, very nice of you to say :)

[@kcodynowahora7245]

this inspired me to finish my linear algebra pset -- awesome content

[@RajKaramchedu]

This video is an eye opener. Back in the day I built Reed-Solomon encoder and decoders and struggled to get the key ideas of Galois theory. I didn’t understand it. Now I am feeling hopeful with your video. I must understand this so I hope you will make follow up videos on this topic. Thank you, thank you!

[@martintrifonov]

I must admit I have no clue what Reed-Solomon encoders are, but it is intriguing to hear they have something to do with Galois theory. Might look into that. Thank you so much for your kind words, I'm glad you enjoyed it!

[@diribigal]

I literally just finished watching through Borcherds' playlist on Galois Theory (mostly review of stuff I learned ages ago), but am excited to complement it with your more-motivated presentation

[@martintrifonov]

I hope you enjoy it!

[@LowHang-nFruits]

so fun to watch

[@Ceaser_exe]

ayyy, this is pretty gooodddd

[@joeaverage8329]

I was studying Harold's Galois Theory book and had difficulty understanding a chapter after this. (Starting on Lagrange resolvent) Great timing for me! Hope you complete this video series. Appreciate your work. :)

[@martintrifonov]

Thanks, Im glad it helped! The Lagrange Resolvent chapter was very illuminating, that’s where the book really takes off :)

[@pandavroomvroom]

perfectly placed

[@johnjakson444]

I had an interesting career in ASIC and DSP design, in one chip it was based on Galois fields and Golay error correction, another was based on the Fast Wavelet Transform with DCT inside and an FFT engine, another was more mundane, building a pattern waveform generator based on sine waves but not using a high resolution Sin ROM, instead computing Sin from low resolution Sin and Cos. Math and ASIC design is a special pairing made in heaven, lots of flexibility to produce the desired result, it's my favourite type of chip design, no arguments about architecture, the math defines tjhe that. Do I remember much about Galois fields, nope, not a darn thing.

[@harriehausenman8623]

Fatnastic stuff! I love the way you explain and summarize. The positively-biased board (black-on-white) really helps me a lot. The audio could get a little bit better, but hey, couldn't it always 😆 Thanks for the ride 🤗

[@martintrifonov]

Thanks, that’s very kind :) will work on the audio!

[@edwardmartin243]

The root structures of 2, 3, and 4 degree polynomials are as follows: 2 deg. x = m + n x = m - n 3 deg. i = (-1)^(1/2) w = [-1 + i(3)^(1/2)]/2, w^2 = [-1 - i(3)^(1/2)]/2 x = m + n + v x = m + wn + (w^2)v x = m + (w^2)n + wv 4 deg. i = (-1)^(1/2) x = m + n + v x = m + n - v x = m - n + iu x = m - n - iu

[@TheMemesofDestruction]

9:04 - Groovy! ^.^

[@prependedprepended6606]

@Martin Trifonov 00:28" Maybe I'm nit picking but shouldn't that be (3^2 + 4 * 2 * 5) under the radical/square root notation as 'c' is -5? Other than that, your presentation was fine and illuminating.

[@Neptoid]

Nice vfx. You have screen recorded a digital art program. I can see the artifacts of the pressure sensitivity graph in the writing. I think the canvas in the video is a 3D asset, but you could have placed it and distorted it with a perspective grid. You are using a soft brush that adds up on overlaps making it look very realistic as if the liquid collected. What blending mode are you using? Is it just opacity? I am guessing you have some textures here and there or some glossy shading applied to the 3D canvas. It looked very good, I might have to steal some of your methods. The editing where the pacing of the sped up writing following you sentences was very good. Sometimes it was in parallel, but that’s what you gotta do no? I think I will use motion graphics with custom typography with this kind of pacing. It is taking a long time to develop, especially when I want to continue to multiply my tasks with pairings between glyphs and innovative characters and transitions, but hey it could become a big animated typeface for math with special dynamic customizable characters! I don’t have much experience with coding so many variables though, but I am beginning to design a custom encoding and variable axes for it… just pseudo code at this point

[@martintrifonov]

Nice detective work :) ! Pretty close, a few things I did a bit different. I didn't screen-record the whiteboard, rather I just made drawings on my Remarkable (E-book type) tablet. I downloaded the drawing files from the tablet to my computer, and I wrote a little code that reconstructs the paths of the lines and animates them. The drawing is reconstructed from the paths using the p5js and p5brush libraries. The latter, p5brush, creates realistic looking brush strokes. And yes, from there on you are mostly correct - some shading and composition! The editing (speeding up the pace of the drawing) was rather labor-intensive. I'm thinking this could be automated a bit in the future. Interesting, what you're describing about a dynamic animated math typeface - but I'm not sure I follow completely. Could you elaborate a bit on what this project is?

[@Alpasonic]

Masterpiece in all aspects - title, artful ambient space composition, scrupulous deliberate manner of presentation, deliberately stylish outfit and haircut (English artistic sophistication a la Oscar Wild? 😅 ),.. and of course fine math

[@martintrifonov]

Such a kind comment, you're being very nice - thank you :)

[@jaddaj5881]

Was Galois aware of the fundamental theorem of symmetric polynomials when he proved his theorem or did he also develop the fundamental theorem of symmetric polynomials along the way to proving there’s no solution by radicals for polynomials of degree 5 or higher?

[@uubuuh]

sub. i plan to someday really understand Galois theory. your video helps.

[@martintrifonov]

Glad to hear it, thanks! :)

[@vemarj2802]

Suggestion: check out the semirings of polynomials, they're pretty cool too.

[@martintrifonov]

Will do!

[@jonetyson]

What video software are you using? It is excellent.

[@martintrifonov]

Thanks! Mostly DaVinci Resolve, and additionally I wrote some code that allows me to create the whiteboard animations easily

[@EdwinSteiner]

Very well explained introduction to a fascinating but quite opaque subject! Great work! I am keenly looking forward to follow-up videos. What you say in the final segment is very true. In particular the Bourbaki collective has killed the human and historical element in teaching mathematics and the tone they set has made modern mathematics somewhat cleaner perhaps but much more difficult, and unnecessarily difficult, to learn. Therefore, pedagogical videos like yours, which teach mathematics in a language and from a perspective more suited to the human brain, are very important. You could improve the audio quite a bit by suppressing reflections in the room you are recording in. It sounds very echoey. (Not like the echo in a large hall or outside, but lots of very fast reflections from nearby surfaces. This is one of the main reasons for bad quality of voice recordings and it becomes very obvious once you hear examples with and without those reflections.)

[@martintrifonov]

Thank you for your kind words and detailed feedback! I actually haven’t heard about the Bourbaki collective before, I’m intrigued! I’m just starting out and I really appreciate the technical feedback on the audio as well :)

[@EdwinSteiner]

@@martintrifonov AFAIK Bourbaki did important work in cleaning up the foundations of algebraic geometry, which had previously gotten into quite a mess and even produced some wrong results due to a lack of rigor. The Bourbaki books were great as an underlying structure but their huge success had a detrimental effect on the pedagogy of mathematics in my opinion, and not only in my opinion, as you can read in the Wikipedia entry: >>As Cartier remarked, "The misunderstanding was that many people thought it should be taught the way it was written in the books. You can think of the first books of Bourbaki as an encyclopedia of mathematics... If you consider it as a textbook, it's a disaster."

[@martintrifonov]

Really appreciate this reply! As for your remarks on the bourbaki collective, this sent me down a rabbit hole of Wikipedia today, can’t believe I never heard about any of this before! As for the audio quality, I actually have a decent microphone (or at least not the cheapest option 😂), but I see I haven’t set it up properly. It was obvious to me that the quality was lacking, and I played around with some noise reduction settings - with limited success. But now I realize much more concretely where and how I could improve. Thanks for your advice :)

[@Vaaaaadim]

If we are given a polynomial of degree 3, with roots a,b,c... can we compute something like sqrt(a) + sqrt(b) + sqrt(c)? Also, can we make use of computing symmetric expressions to give us good starting points for polynomial root finding algorithms or modify the iteration steps in one? I'm imagining for example... unlikely scenario but, if you knew for example that your given polynomial has only real roots, and one root is particularly larger than the others, you could approximate that large root by computing (r1^50 + r2^50 + ... + r_n^50)^(1/50).

[@martintrifonov]

Im not sure I have a good answer to this - but what you are proposing loosely reminds me of Graeffe‘s Root Squaring Method - might be worth looking into :)

[@Vaaaaadim]

@@martintrifonov Looking at this again with fresher eyes, at least for the first question I asked I think the sums of powers of roots can be applied to taylor series. For sqrt(x) a taylor series centered at x=1 converges if x is in [0,2]. If we only have real, positive roots we could compute sqrt(r1) + sqrt(r2) + ... + sqrt(r_n) by squishing the roots closer to the origin by substituting cx for x in our starting polynomial, where we can pick a constant c. After computing the taylor series to an accuracy we like, we can multiply the sum of those approximations by sqrt(c) to get the final answer. It should be relatively straightforward to determine the largest magnitude a root might have and pick c to be that largest magnitude.

[@jbman890]

sorry if dumb, but am I correct in saying (xy)^4 + (zy)^4 + (xz)^4 is a polynomial? and cannot be said to be represented using the elementary polynomials according to the proof, because the table doesn't have any entries of the form (xy)^n?

[@martintrifonov]

It's a fair question! It is a symmetric polynomial, yes - and you're right to say that we can't learn how to represent it using using only elementary symmetric polynomials by the argument in the first proof. The first proof only shows how to represent power sum polynomials (x^k+y^k+z^k) using elemetnary symmetric polynomials. However, the second proof in the video shows how to do this with any symmetric polynomial, including your example. Hope that makes sense!

[@jbman890]

@@martintrifonov thanks, should have finished the video first

[@jbman890]

@@martintrifonov Just finished the video, I see that the polynomial I suggested can be expressed as ((s4)^2 -s^8)/2. Thanks for the knowledge

[@WelshPortato]

good maths and serious drip

[@arekkrolak6320]

How did you move from quadratic to quintic without explainig cubic and quartic? :)

[@martintrifonov]

If this becomes a series the quintic would be the next chapter! :)

[@ojas3464]

👍

[@danielshook2442]

Great video! My only suggestion would be a differently colored background as your hair blends in and is a bit distracting. Looking forward to more.

[@martintrifonov]

You're right, that didn't quite turn out how I hoped it would. Will be working on the production quality (the audio too) in future videos. Thanks for the feedback!

[@BlackCodeMath]

If Saturday Night Live did a skit on math I am trying to learn.

[@tasteful_cartoon]

MMM is a catchy intro. The effort put into this video clearly shows, kudos! the proof at 15:30 wasn't crystal clear to me. can you expand (or guide me to a resource) on how this table layout proves Newton identities on the general case?

[@martintrifonov]

Thanks! If I find time I will make a more formal write up of the contents of the video. The manner in which I present the proof (using the table) I haven’t found anywhere else - however, it is, in essence, very similar to one possible derivation given in the Wikipedia page of Newtons Identities. If you go to the page for newtons identities, and navigate to the subtitle „Derivations > As a telescopic sum of symmetric function identities“ you can read more there.

[@csaracho2009]

I am here because the whiteboard is magical... writes himself on!

[@mg9581]

I hope your middle name is Sam or something so you can start making videos on Saturday/Sunday.

[@SigmaChuck]

Would it be rude to point out that his solution for the roots of the quadratic is wrong?

[@martintrifonov]

Which solution is that?

[@SigmaChuck]

0:34 @@martintrifonov

[@mestar12345]

the one that says -4 x 2 x 5, and should say -4 x 2 x (-5) @@martintrifonov

[@martintrifonov]

Right, I see. Oops! Thanks for pointing it out! Ill add it to the video description.

[@alexander_adnan]

Bob ross style 😂😂😂😂 nice ❤❤❤❤❤

[@MathsSciencePhilosophy]

I wish I had a whiteboard who could understand what I wanna write on it just like your whiteboard 😂

[@Etothe2iPi]

0:35 Wrong sign under the root.

[@martintrifonov]

Oops. My mistake. Thanks for pointing it out!

[@getoffmeow]

that drip

[@usercommon1]

zamn

[@megadeth205]

Мавроди?

[@user-zu6tz9fy2r]

Improve audio quality please

[@martintrifonov]

Working on it!

[@DDranks]

kzbin.info/www/bejne/aZrQloedfKeUgsUsi=i7s_yO2lWOolAVQE&t=192 I know it's a famous fact that you can factorise a polynomial like this, but I think you should either justify it, or at least tell a name of the theorem, so it'd be easily googleable, as you do just 20 seconds earlier with the fundamental theorem of algebra.

[@martintrifonov]

You’re right, I should have! Thanks for the feedback, I appreciate it :)

[@vNCAwizard]

Audio quality is not great. That makes it hard to focus upon your lecture, which I want to hear. You will well serve your viewers by attending this niggling issue.

[@martintrifonov]

Noted, working on it. Thanks for the feedback!

[@dominiquelaurain6427]

I had the same obstruction for the Galois theory : too much theory to learn before solving any equation....it is the same than a promise for a good dessert but you have to eat a lot of heavy dishes ;-) For me Lagrangian resolvents was the point I stopped, because despite I understand the main ideas (splitting fields and so on, .... more in the group theory and fields) the difficulty is to guess whether a given equation is Galois solvable and how to process in that case. There is an unknown too : can you apply the theory when the coefficients ARE NOT rationals (not integer).. is this for "algebraic equations" only ? Guessing the resolvents (or the intermediary fields) is the extra I need ....the new entries in your "bag".

[@martintrifonov]

I like your analogy with dessert - it does feel like that sometimes :)

[@briansmith7458]

Made it to 10:22 then I got lost.

[@martintrifonov]

That's valuable feedback, it's good to know which moments might be hard to follow, so thanks!

[@Zaid-mw2mq]

Hogwarts math!

[@Oysters176]

I am in love with you. And I am a guy.. I am questioning my sexuality..

[@user-sk4kg4hr3k]

Sound is pretty low quality. Also bad diction/unclear pronunciation. Content is ok

[@dickinaround87]

Martin - GREAT video. You have a fantastic ability to explain these concepts with very helpful visuals. One suggestion though - use a lapel mic when talking in front of the whiteboard! You've earned a subscribe from me!

[@martintrifonov]

Thank you, it means a lot :) And also for the advice, I'll definitely work on the audio quality in future videos!