# Galois Theory

First off, I’m spreading posts out from now on. I just don’t have the time to push out this much math when I’m supposed to be doing other math. So what is Galois Theory? The idea is to look at fields and and how fields contain each other, and relate them to groups and how groups contain each other. This will (though it may seem odd) depend largely on roots of polynomials. This is good though, because it will allow us to answer questions about polynomials (like the existence of a quintic equation). There are a few things you should be comfortable with to really attack Galois Theory

So let’s get started. In general, we’ll want to work in algebraically closed fields. An algebraically closed field is one in which every polynomial with can be factored into linear terms. If you know the fundamental theorem of algebra, another way to state it is that $\mathbb C$ is algebraically closed. An example of a field that is not algebraically closed is $\mathbb Q$. The polynomial $x^2-2$ cannot be factored over $\mathbb Q$.
I’m not sure if I’ve used the notation before, but if $k$ is a field, then $k[x]$ is the polynomial ring over $k$. If you don’t know what a ring is, no sweat. It’s basically a field that you can’t always divide in. But that’s not even so important. The important part is that $k[x]$ just means “all the polynomials” whose coefficients come from $k$.
We can use this notation to state algebraic closure in a different way. If $k$ is algebraically closed if $p\in k[x]$, there exists some $\alpha\in k$ for which $p(\alpha)=0$. Notice that I just said $p$ has one root, but before I required it to have all the roots. That’s fine. If it has one root, I can factor out that root. By the algebraic closure of  $k$, I can find another root. I can keep doing this until I have factored it into linear terms.