What are...the groups of Galois theory?

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Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.

This time.
What are...the groups of Galois theory? Or: An ocean of symmetric groups.

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Slides.

TeX files for the presentation.

Thumbnail.

Main discussion.

Background material.

Computer talk.

Pictures used.
Pictures created using Mathematica’s Roots
Pictures created using Mathematica’s MatrixPlot
More pictures created using Mathematica’s MatrixPlot
Another picture created using Mathematica’s Roots

YouTube and co.

#algebra
#grouptheory
#mathematics

VisualMath

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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!

yankeed

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.

tuongnguyen

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

jasonpalin

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

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

eternaldoorman

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

Jaylooker

What are...the groups of Galois theory?

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What are...the groups of Galois theory?

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