Galois group

Imagine you have a bunch of different things (mathematical objects) and you want to see how they're related to each other. One way to do this is to group them together based on certain similarities or differences they have.

The Galois group is like a special kind of grouping that we use in math to understand how different solutions to equations are related to each other.

Let's say you have an equation like x^2 - 5 = 0. This equation has two solutions: x = sqrt(5) and x = -sqrt(5). But how are these solutions related to each other? That's where the Galois group comes in.

We can think of the Galois group as a group of "symmetries" that "act" on the solutions of the equation. In this case, the Galois group for the equation x^2 - 5 = 0 is a group of two symmetries: the identity symmetry (which does nothing), and a reflection symmetry that switches the two solutions (like flipping a coin).

If we have another equation with different solutions, it might have a different Galois group with different symmetries. By understanding the Galois group for an equation, we can learn more about its solutions and how they're related to each other.

So the Galois group is like a secret group of symmetries that helps us understand how all the solutions to an equation are related to each other.