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

[hydose]

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

[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.

[sydneythesurfboards]

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! 🇸🇪

[Ricocossa]

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! 😊

[videos-de-fisica]

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

[nickush]

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.

[joevogl]

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

[glumbortango]

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.

[MrSamwise]

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!

[clickaccept]

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

[jlas]

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!!

[mohammadareeb]

Awesome video!! Would love to see a follow up

[tempiadem]

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!

[vladthemagnificent]

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

[swapnilshrivastava]

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...❤

[christiansmakingmusic]

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.

[imrematajz]

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!

[willyballmann]

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

[grinchsimulated]

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

[robharwood]

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!