Let $E$ be an elliptic curve over a number field $k$. By definition, $E$ has a distinguished $k$-rational point, we denote it by $\mathcal{O}$. Given any other closed point $P \in E$, can we find an automorphism $\tau: E \rightarrow E$ such that $\tau(P) = \mathcal{O}$?
Lemma 4.2 in Hartshorne's Algebraic Geometry says this is possible but I think it is under the assumption that $k$ is algebraically closed (and so closed points are $k$-rational). By the answer to this post, for such an automorphism to exist we also require that the residue field of $P$ and $\mathcal{O}$ are equal. This generally isn't true since $\kappa(P)$ is a finite extension of $k$ of degree $\geq 1$.
So my question should be: does this mean that Lemma 4.2 will fail to hold if $k$ is a number field?