TITLE: $\mathrm{Aut}(\mathbb{Q}(\pi)/\mathbb{Q})=$?
QUESTION [7 upvotes]: Perhaps a silly question. I'm trying to understand trascendental field extensions, but I can't find a lot of instructive examples.

Consider the extension $\mathbb{Q}(\pi)/\mathbb{Q}$. What is its group of automorphism, $\mathrm{Aut}(\mathbb{Q}(\pi)/\mathbb{Q})$?

REPLY [4 votes]: If $F$ is a field and $K = F(\alpha)$ is a transcendental extension, then every $F$-automorphism of $K$ is of the form
$$\alpha \mapsto \frac{a\alpha + b}{c\alpha + d},$$
where $a,b,c,d \in F$ with $ad - bc \neq 0$.