# The group of field automorphisms

Recall that a field automorphism is a function that “respects all of the properties of fields.” For instance, if $f$ is a field automorphism, then $\phi(0)=0$, $\phi(1)=1$, $\phi(x+y)=\phi(x)+\phi(y)$, and $\phi (x y)=\phi(x)\phi(y)$.

I did leave out one rule of note $\phi(x^{-1})=\phi(x)^{-1}$. Why did I leave this one out? Because I can prove it from the other ones…

$1=\phi(1)=\phi(x\cdot x^{-1})=\phi(x)\phi(x^{-1})$,

so $\phi(x)^{-1}=\phi(x^{-1})$. So basically, anything remotely reasonable I can say about fields passes through the automorphism.

If I were to write down all of the automorphisms of a given field, what could I say about them? As I mentioned last time, there is always the trivial (a.k.a. lame) automorphism that doesn’t do anything. You plug in $x$, it gives you back $x$ exactly as is. We call this the identity automorphism. In general in mathematics, “identity” can be loosely translated as “lame.”

If $\phi$ is an automorphism on some field $F$, we required it to be a bijection (one-to-one and onto). So it has an inverse, which we will write as $\phi^{-1}$. This is a field automorphism as well. This one simply undoes whatever $\phi$ did.

If I have two field automorphisms $\phi$ and $\psi$, then I can compose them. You can verify that this is also an automorphism. I’ll show that it respects addition. The rest is just as easy.

$\psi(\phi(x+y))=\psi(\phi(x)+\phi(y))=\psi(\phi(x))+\psi(\phi(y))$

So we can compose automorphisms and take inverses. Oh, and there is an identity automorphism. Sound familiar? Such objects are called groups. It turns out, whatever structure we are talking about, the collection of automorphisms form a group. This will be an extremely important tool, so we should give it a name. We’ll denote the group of automorphisms of a field $F$ by $\mbox{Aut}(F)$.