My problem is the following:
Let $K, L$ be fields and $u_1,\cdots, u_n$ be $n$ distinct field homomorphisms $K\to L$, define $E:=\{a\in K: u_1(a)=\cdots=u_n(a)\}$. Then $[K:E]\geqslant n$.
One can view $K$ as a subfield of $L$ via $u_1$ and assume $u_1={\rm id}$ and by considering normal closure of the image of the $u_i$'s inside an algebraic closure of $L$, one can assume the extension $L/K$ is normal, and each $u_i$ extends to an automomorphism $v_i$ of $L$. I want to use the group $H$ generated by the $v_i$'s (so $E=K\cap L^H$) and apply Artin’s lemma saying that if $H$ is finite, then $L/L^H$ is a finite Galois extension with group $H$. But I'm afraid $H$ can be in finite. Any ideas?