Galois fields are used in crypto a bit, and I recommend this lecture to understand it in the context of crypto

https://www.youtube.com/watch?v=x1v2tX4_dkQ

PS: He has an entire series of lectures on his channel. Highly recommend.

Fun fact: Evariste Galois made major contributions to math in his teens, before dying in a duel at age 20.

https://en.wikipedia.org/wiki/%C3%89variste_Galois

This document was also mentioned yesterday on the 'Why is the quintic unsolvable?'[1] post. The post also has some additional interesting references in the comments. Worth checking out!

[1] https://news.ycombinator.com/item?id=14685466

Can anyone help me understand what is happening at the bottom of page 23 (page 3 of the PDF)?

It says any permutation sigma of x_1, ..., x_n can be extended to a bijection of Q(x_1, ..., x_n) defined by

    sigma f(x_1, ..., x_n) = f(sigma x_1, ..., sigma x_n).

But I don't see how this definition can be consistent. For example, let

    f(a, b) = a - b
    g(a, b) = a/a + b/b = 2

    x_1 = 5
    x_2 = 3
    sigma x_1 = x_2
    sigma x_2 = x_1

Then according to the formula:

    sigma f(5, 3) = sigma (5 - 3) = sigma 2 = f(3, 5) = -2

But

    sigma g(5, 3) = sigma 2 = g(3, 5) = 2

Contradiction?

Galois had an insight that I always seemed particularly deep to me: that problems should be classified not by topic area (analysis, theory of equations, geometry) but by their underlying form.

Galois was ahead of his time.[1]

[1] You could say he was ahead by a century, to quote a famous song.

Beginners?

Galois theory without functor... Without fundamental theorem of algebra. Are you kidding?

I have never studied group theory, so this is way beyond what I'm ready for. But I scanned over a few sections and read to the point that I got lost.

The part that surprised me is that it seems to focus on rational numbers. I always assumed that Galois Groups were focused on more abstract concepts of sets. Is Galois Theory mostly about rational number (or even real numbers), or is the author just using the rationals to keep the paper focused on beginners?

this guy was AMAZING at maths