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. __ 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. (https://www.youtube.com/playlist?list=PL8yHsr3EFj53Zxu3iRGMYL_89GDMvdkgt) (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: https://www.youtube.com/playlist?list=PLelIK3uylPMGzHBuR3hLMHrYfMqWWsmx5) 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 Follow me! Twitter: @00aleph00 Instagram: @00aleph00 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)