Why is every monomorphism of E into $\overline{\mathbb{Q}}$ a $\mathbb Q$-monomorphism of E into $\overline{\mathbb{Q}}$?

Question

I've been trying to solve the problem below, but I'm not even sure how to get started. Any help would be greatly appreciated. I feel like there is a key insight that will solve the problem, but I'm not seeing the insight.

Let $E$ be a field containing $\mathbb Q$ and let $\sigma : E \rightarrow \overline{\mathbb{Q}}$ be a monomorphism. Show that $\sigma(a) = a$ for all $a \in \mathbb Q$.

Answer 1

Hint: Where does $\sigma$ send 1, 2, and 1/2?