Let $Aut(\mathbb{C}/\mathbb{Q})$ be the set of field automorphisms of $\mathbb{C}$ over $\mathbb{Q}$ (in short, all field automorphisms of $\mathbb{C}$). Let $x$ be an element of $\mathbb{C}$ such that the set $\{\sigma(x)|\sigma \in Aut(\mathbb{C}/\mathbb{Q})\}$ is finite. Is it true then that $x$ is algebraic over $\mathbb{Q}$ (and if so, why?) ?
It's being used in the following paper about elliptic curves to prove that a elliptic curve with complex multiplication has modular invariant $j$ which is algebraic over $\mathbb{Q}$: http://www.math.tifr.res.in/~eghate/cm.pdf
Any help would be appreciated.