I am thinking about $\mathbb{C}/\mathbb{Q}$. But other than the identity map and taking complex conjugates, I cannot think of anything. Any ideas? Thanks.
Edit I just managed to show that if $L/K$ is algebraic then $\theta$ must be injective. For if $\theta(a)=0$ then so must $\theta(p(a))=0$ where $p(X)$ is the minimal polynomial of $a$ over $K$. But $\theta(p(a))=p(\theta(0))\neq0$ because the minimal polynomial cannot have zero constant term. And if this extension is finite that proves it is an isomorphism. So now I am asking if we can be more ambitious than just requiring finite extensions--what if $L/K$ is only assumed algebraic? And what if we drop the assumption that $L/K$ is algebraic?