finite extension $K\subseteq L$ whose $\mathrm{Aut}(L/K)$ is infinite group?

Question

Is there any example of finite extension of field $K\subseteq L$ whose $\mathrm{Aut}(L/K)$ is infinite group?

Answer 1

No. A finite extension is algebraic, so each element of a basis of $L$ over $K$ can go to only finitely many places under a $K$-automorphism of $L$. A $K$-automorphism of $L$ is determined by where it sends the elements of a $K$-basis of $L$, so there are only finitely many $K$-automorphisms of $L$.

Answer 2

No, for every finite extension $L/K$ the group of automorphisms of $L$ over $K$ has cardinality at most $[L\colon K]$.

However, there exist infinite dimensional algebraic extensions with trivial group of automorphisms, for example $\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{3},\ldots)/\mathbb{Q}$