What are...the groups of Galois theory?

[@yankeed4793]

Galois theory - something about unsolvability of quintic polynomials, or so I’m always told, followed by a stern warning that Galois theory is really hard - not really very satisfying. What can Daniel do in 16 minutes to make this interesting, let alone “fun”? 16 minutes and 11 seconds later - quite a lot it turns out! Yes, who knew Galois theory can be funny! Love how you convey the key interesting insights in a few minutes without overly dumbing down the topic. You always take pains to note whenever you are being a bit loose with the “rigor” to get across what is important, interesting, and yes, funny. Lots of resources out there offering rigorous treatment of the subject, not many that inspire one to take the trouble to do so. I’m unlikely, personally, to dive deeper into Galois theory, but feel like I have a deeper insight into what it is about. (Although I told myself the same thing when you did your topology intros and, like Alice, I got pulled down that rabbit hole ;) Another great favorite theorem addition and keeping it fun!

[@VisualMath]

Rabbit holes are very dangerous - I have been pulled down too many to count ;-) And sorry for the topology rabbit hole ;-) Anyway, thanks for the feedback, as usual. You make a good point: way too often the answer is “its too hard” in math and then, surprise surprise, people get turned down. We as a community really need to work on this front - math is fun and full of rabbit holes, so why not make that story more popular?

[@tuongnguyen9391]

I have not been able to watch the whole video but to be admit I really like your style. As an engineer I and my colleague really need to quickly zoom in on the application and physical interpretation of any mathematical concept.

[@VisualMath]

Thank you for the feedback - that is very much appreciated. Sorry that you didn't make it to the end - I hope it was not too boring! (Totally to be blamed on me.) I think you are following a fantastic strategy and I hope the video was somewhat helpful on your way.

[@jasonpalin4521]

Vaguely relating to symmetry groups - Tits buildings would be a cool topic for you to make a video about.

[@VisualMath]

Ah, how could I forget buildings? A fabulous idea, its on my list now. Thanks!

[@1.4142]

What you do in class: find the zeros of x^2-2x+1 What's on the test: 13:39

[@VisualMath]

I am laughing and crying at the same time - very true ;-)

[@eternaldoorman5228]

These are algebraic numbers that want to be transcendental when they grow up?

[@VisualMath]

Oh, I like that analogy - thanks for sharing. Yes, they work very hard to be transcendental, but do not quite get there ;-) Most algebraic numbers are of that form.

[@Jaylooker]

I wonder if there is some mathematical structure that can express all roots of any polynomials. Maybe the p-adics could be useful.

[@VisualMath]

Hmm, interesting question. Sadly, I do not know too much about this: time to fix this; sounds exciting! - As you say the p-adic numbers come to mind. - What also comes to mind is the nontrivial question how to implement \bar(Q) (so the algebraic numbers) in a computer algebra system.

[@Jaylooker]

@@VisualMath Maybe a certain topos applies? The p-adics come to mind because they can have infinite decimal places to maybe describe e or π. I am taking implementing \bar(Q) of algebraic numbers in a computer algebra system as in asking for roots of polynomials. I found approximation for polynomials can be implemented but that may not being what is wanted.

[@VisualMath]

@@Jaylooker Ahh, that is not quite want I meant. Computer algebra systems do exact calculations. So implementing \bar(Q) is not about approximations (as you point out approximations are "easy to get", and not what we want) but about honest algebraic expressions for the roots. See for example here www.sciencedirect.com/science/article/pii/S0747717109001497 which gives "the illusion of \bar(Q)" (in the words of the author), so gets quite close to an answer to your question. Or rather it gives the illusion of an answer ;-)

[@Jaylooker]

@@VisualMath Got it. The absolute Galois group (\bar(Q)/Q) could be considered though it is difficult to understand. There is a way to get at using Beyli’s theorem and dessins d’enfants. I was also thinking it could be a Reimann-Hilbert problem with the étale fundamental group of the absolute Galois group by way of Grothendieck Galois theory is taken to be the monodromy (fundamental group) with corresponding differential equations. Having it as some fundamental group allows homotopy type theory to apply and how that is implemented on a computer.

[@VisualMath]

@@Jaylooker "Having it as some fundamental group allows homotopy type theory to apply and how that is implemented on a computer." Getting HoTT involved sounds like a fabulous idea. I wonder if someone has done something in this direction.