Hilbert's theorem 90
Hilbert's Theorem 90 is about something called Galois theory, which is just a fancy way of talking about how numbers behave when you do math with them. Specifically, it says that if you have a certain type of number (which we call a "field"), and you raise another number to a certain power (which we call an "automorphism"), then the answer is always 1, no matter what numbers you're using.
Think of it like playing with blocks. You have a bunch of blocks in a big pile, and you want to know what happens when you stack them up in different ways. If you have a certain type of block (which we'll call a "field" block), and you stack another block on top of it (which we'll call an "automorphism" block), then no matter what other blocks you use, the stack will always be the same height as when you started. And the amazing thing is that this works for all different types of blocks and all different kinds of stacks!
To understand why this works, think about how we can take a number and "split" it into two pieces - like breaking a cookie in half. We can then take these pieces and swap them around, like switching the top and bottom halves of a sandwich. This is what we call an automorphism, and it turns out that if we do this with the right types of numbers, the answer will always be 1.
So, Hilbert's Theorem 90 is really just saying that if we have two numbers that multiply to 1 (which is like having two halves of a cookie that add up to the whole), then there's always a way to split them up and swap them around using automorphisms, so that the answer is also 1. And this works not just for cookies or blocks, but for any number we can think of, no matter how big or complicated it might be!
Think of it like playing with blocks. You have a bunch of blocks in a big pile, and you want to know what happens when you stack them up in different ways. If you have a certain type of block (which we'll call a "field" block), and you stack another block on top of it (which we'll call an "automorphism" block), then no matter what other blocks you use, the stack will always be the same height as when you started. And the amazing thing is that this works for all different types of blocks and all different kinds of stacks!
To understand why this works, think about how we can take a number and "split" it into two pieces - like breaking a cookie in half. We can then take these pieces and swap them around, like switching the top and bottom halves of a sandwich. This is what we call an automorphism, and it turns out that if we do this with the right types of numbers, the answer will always be 1.
So, Hilbert's Theorem 90 is really just saying that if we have two numbers that multiply to 1 (which is like having two halves of a cookie that add up to the whole), then there's always a way to split them up and swap them around using automorphisms, so that the answer is also 1. And this works not just for cookies or blocks, but for any number we can think of, no matter how big or complicated it might be!