I currently have to cope with field automorphisms. I already understood that any field automorphism of $\mathbb{C}$ must fix all elements in $\mathbb{Q}$.
My question is the following: Assume a number $x$ is fixed by every field automorphism of $\mathbb{C}$. Does that imply that $x \in \mathbb{Q}$?
My strong guess is that this implication is true but I could not come up with an idea for a proof. Any hint into the right direction would be appreciated.
It turns out to be true. Since I don't think your question is a duplicate, but it is nicely addressed at this solution by Andres Caicedo, I'm giving you a community wiki answer to point you to it.
If you take a look at the discussion right before the second to last paragraph, you can learn why exactly the rationals are fixed.