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
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! 😊
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.
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!
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.
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! 🇸🇪
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!
It took me around 3 nights to watch the video (as I watch RU-vid 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!
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.
Cannot wait for the next part.. This has been an elusive topic for me and for the first time ever it has made any sense to me after watching this video. I had to subscribe immediately.. Please keep making more of these...❤
Very nice, Martin. Hope you make more of these. Galois Theory is beautiful but not as mystical as it gets painted by some. Permutations of roots in the splitting fields loses its mystery when you think of it as simply flipping radicals within dense subsets of the real numbers, a+root(2)b, and a-root(2)b are just reflections of each other. If the other roots of Unity are involved a similar abstract picture emerges, another axis with orderable values which get permuted. I know there’s more to it but this is essentially what it is. Would love to see you take this series all the way through the Galois results.
Great content. Thank you. I'm just stepping into this level of math, and you have deepened my understanding. There is an annoying echo on the audio though. I don't mind replaying parts to be sure of understanding the math; but having to replay six times just to catch the word "norm" was truly annoying.
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
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!!
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 :)
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.
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!
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!
ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-3imeTgGBaLc.htmlsi=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.
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
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
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!
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.
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?
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
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?
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 🤗
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. :)
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!
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!
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
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.)
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 :)
@@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."
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 :)
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).
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 :)
@@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.
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?
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!
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!
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".
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?
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.
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!
