Local-global test of algebraicity Let $\alpha \in \mathbb{C}$. Suppose for all primes $p$ and all isomorphisms $j : \mathbb{C} \rightarrow \mathbb{C}_p$, $j(\alpha) \in \bar{\mathbb{Q}}_p$. Is $\alpha \in \bar{\mathbb{Q}}$?

Sure, if $\alpha$ is not algebraic take $\beta\in \Bbb{C}_p,\not \in \overline{\Bbb{Q}}_p$ and $\sigma\in Aut(\Bbb{C}), \sigma(\alpha)=j^{-1}(\beta)$.
(those things require axiom of choice)
