What is the square root of two? | The Fundamental Theorem of Galois Theory

This video is an introduction to Galois Theory, which spells out a beautiful correspondence between fields and their symmetry groups.

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SOURCES and REFERENCES for Further Reading!

This video is a quick-and-dirty introduction to Galois theory. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below.

(a) Galois Theory

Galois Theory notes by Tom Leinster: These notes are by far the best resource out there for learning the subject. They’re completely rigorous, but they’re also written in a very reader-friendly way with lots of examples and motivation. (See link here: https://www.maths.ed.ac.uk/~tl/gt/gt.pdf)

This playlist on Galois Theory by Professor Richard Borcherds is a gem. It explains Galois Theory from the ground up, rigorously, in almost complete generality. (   • Galois theory  )

(b) Group Theory

Group Theory lectures: This playlist by Professor Benedict Gross is a beauty. It goes through the entire group theory syllabus from the ground up, and Professor Gross is a masterful lecturer. (see link here:    • Abstract Algebra  )

MUSIC CREDITS:
The song is called “Taking Flight”, by Vince Rubinetti.
https://www.vincentrubinetti.com/

THANK YOUs:

Extra special thanks to Davide Radaelli and Grant Sanderson for helpful conversations while making this video!

SOFTWARE USED:
Adobe Premiere Pro for Editing

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Intro: (0:00)
What is the square root of 2?: (1:08)
Fields and Automorphisms: (6:04)
Examples: (8:55)
Group Theory: (16:34)
The Fundamental Theorem: (18:25)