# Fixed fields

If I have a field $k$ and a field automorphism $\sigma$, I can ask about the fixed points of $\sigma$. That is, all of the $x\in k$ such that $\sigma(x)=x$. Notice that if $\sigma$ fixes $x$ and $y$, then $\sigma(x+y)=\sigma(x)+\sigma(y)=x+y$ and $\sigma(xy)=\sigma(x)\sigma(y)=xy$, so $\sigma$ fixes $x+y$ and $xy$. In fact, the set of fixed points of $\sigma$ is itself a field! I just checked two of the rules. You can check the rest.

This field is called the fixed field of $\sigma$, and it’s super super important. If we have two automorphisms, $\sigma$ and $\tau$, we can take the intersection of their fixed fields. The intersection of fields is always a field. What field is it? It’s the field consisting of everything fixed by both $\sigma$ and $\tau$.

But why stop at two automorphisms? What if I take a group $G$ of automorphisms of $k$. That has a fixed field. It’s called the fixed field of $G$. I could take the entire automorphism group of $k$, or I could just take a subgroup of it. The bigger the group I take, the more fields I intersect, so the smaller my fixed field is.

That’s an important idea. Big group means small fixed field. Small group means big fixed field. We’ll make this precise very soon.