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, $\mathrm{Aut}(\mathbb{Q}(\pi)/\mathbb{Q})$?
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 $ad - bc \neq 0$.