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

(b) Group Theory

MUSIC CREDITS:
The song is called “Taking Flight”, by Vince Rubinetti.

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)

[rickdoesmath]

You just made me, an analysis lover, watch 25 minutes of abstract algebra content. That is incredible

[JimFarrand]

This video is a thing of genuine beauty. You have a rare talent for illuminating these deeply technical subjects in a fascinating and accessible way. Many thanks.

[Acryte]

Took group theory and ring and field theory but didn't get all the way to Galois theory by the end. What a clear and concise way to encapsulate the fundamental concept. Thanks for this.

[Archer-bccv]

Thank you for putting so much effort into making this. This is my first time hearing about Galois Theory, and this video was amazingly clear and a treat to watch. It's sad that so few people watch these compared to other channels of equal quality.

[soumyasarkar]

It's a testament to the complexity of groups and Galois Theory that simplified explanations still manage to fly over your head, but equally it is a testament to the beauty of these concepts that every time you want to go through it once again simply to understand more. This was a fantastic video - probably the most beginner friendly of all the videos I saw in this area!

[jogloran]

Huge fan of your explanation style and visuals! Can't wait to watch this.

[paulmeixner]

I find it difficult to express just how GOOD this video was at explaining the general idea behind Galois theory. Genuinely, thank you. Thank you so much, you've given me another way to look at fields. Another tool that I didn't know even existed.

[logiclrd]

When you said that the lines wouldn't go where I expected, I almost paused the video, because I was pretty sure I did see where they would go -- and I was right! My intuition was based on the understanding of multiplication by any complex number of magnitude 1 as rotation -- which of course wraps around after each full turn. So, ζ⟶ζ² applied twice is just the rotation by ζ² twice -- ζ² * ζ² = ζ⁴. Well, that's reasonably obvious, but the next step falls out of the wrapping nature of the rotation. ζ⁴ twice is just ζ⁸, but since ζ is the 5th root of unity, that means that ζ⁸ = ζ³ -- and so on. With this view, all of the graphs are immediately evident from the starting point of the given mapping of ζ.

[MegaBruceh]

This is the most perfect introduction to Galois Theory that I have seen over several decades. It gives us not just the bare bones of the theory, but also their subtlety, their power, and beauty, and every idea copiously illustrated by clear diagrams and algebraic formulae. However, there must be something wrong with my pc, or my old ears, as I can hardly hear the voice over the music. I wish I could turn the music down, down to zero, and then I would enjoy the video for its full worth!

[karltraunmuller]

I loved this stuff so much when I was a young computer science student. Finite fields, coding theory, polynomials. Heck yes.

[tracymarcinkos]

Really missed a lot these videos. Thanks for coming back!

[kiezwxf]

Less than 5 minutes in watching and I have made more Google searches than required by an assignment. I love Maths and Engineering.
Keep up the good work.

[p_square]

Finally a video after 9 months! Feels great man

[electra_]

The thing that popped out at me when I finally understood the usefulness of this:
It was not at all clear that Q(zeta) should necessarily contain Q(sqrt(5)).
I can see why such a resemblance might exist given that zeta is the 5th root of unity, but this was not at all obvious.
But, we'd already seen the subgroup of Q(zeta), it pops out very clearly in the table! and this alone is enough to prove the field contains some subfield.

[amorphous_gus]

Never learned galois theory in school, just some basic group theory and field theory. I always imagined it was very daunting, but this 25 minute video was very easy to follow and gives me a sense of why people even care about this field. Thank you

[johnchessant]

The connection to unsolvability of the general quintic:
Suppose x is a root of an irreducible quintic polynomial, and x is expressible by radicals. Then we can "build up" to a field containing x with a sequence of fields like Q < Q(a) < Q(a, b) < Q(a, b, c), where each step we adjoin an nth root of some element of the previous field.* Each step's Galois group will be a cyclic group. Using the Galois correspondence, this means the Galois group of the last field will have the property that it has a sequence of (normal) subgroups where the quotient at each step is a cyclic group. This is what we call a "solvable group".
However, the Galois group of a general quintic polynomial is the symmetric group S5, which does not have this property. When you try to form a sequence of subgroups, you run into the alternating group A5 which doesn't have any nontrivial normal subgroups. Hence we have a contradiction, so the original assumption that x was expressible by radicals is false.

*e.g., say x = sqrt(2)+sqrt(3+sqrt(5)), then we'd do Q < Q(sqrt(2)) < Q(sqrt(2), sqrt(5)) < Q(sqrt(2), sqrt(5), sqrt(3+sqrt(5))) so x is contained in the last field in the sequence. You can do this for any radical expression for x.

[nerdsgalore]

As someone who has no experience in the more abstract side of math, this video was surprisingly clear!

[singtatsucgc]

If there had been YouTube in my teens, I would have studied math at college. Thanks for posting! I find this enormously interesting and satisfying to watch in my middle age.

[ivanklimov]

this channel is extremely underrated, some of the best math content on youtube. no other vid has ever gave me as good of an intuition for this topic, and i've seen a lot of them

[Jason]

This is amazing! I have had a difficult time trying to find a good explanation of Galois theory, and this finally made it click. Thank you so much!