Group scheme of roots of unity

Hey kiddo! Have you ever heard of roots of unity? They are special numbers that when you raise them to a certain power, they equal one. For example, let's say we have the number 1. If we raise 1 to the second power (1^2), it still equals 1. If we raise 1 to the third power (1^3), it still equals 1. And if we keep going, we'll find that no matter how many times we raise 1 to a power, it will always equal 1.

Now imagine we have other numbers like 2, 3, 4, and so on. If we raise those numbers to a power, they won't always equal 1. But there are some special numbers (just like 1) that when raised to a power, they will always equal 1. Those numbers are called roots of unity.

Now let's talk about a group scheme of roots of unity. A group is like a club of numbers that have certain rules for how they can interact. In this club, all the members are roots of unity. And the scheme part means that there's a special way that we can put these roots of unity together so they form a pattern.

For example, let's say we want to make a group scheme of roots of unity where each member is raised to the fourth power, and they all equal one. We would start with the number 1, then find the fourth root of unity (which happens to be -1) and add it to our club. Next, we would take the square root of the fourth root of unity, and add that to our club (which is i). Finally, we would take the square root of i (which is the fourth root of unity), and add that to our club (which is -i).

Now we have a group scheme of four roots of unity that all equal one when raised to the fourth power. And we can use this club to solve math problems or do other cool things in math. Pretty cool, right?