I know that the easiest way to show a point is not definable is to find an automorphism of the structure that moves the given point. I've also seen many examples undefinable points that couldn't be moved by any automorphism of the structure. In tackling this problem, I've been trying to construct an arbitrary elementary extension of any structure, $\mathfrak A$, and then showing that there is an automorphism of that structure that moves the point. However, I've had some holes poked in the constructions I've made. If anyone has any advice about my method, or more general knowledge of such a problem, I'd be most appreciative.
Let $\mathfrak A$ be a structure and suppose that $a \in |\mathfrak A|$ is not definable over $\mathfrak A$. Show that there is an elementary extension $\mathfrak B$ of $\mathfrak A$ and an $f \in \operatorname{Aut}(\mathfrak B)$ such that $f(a)\ne a$.