Prove that every field with characteristic 0 contains a countable subfield.
Hint
If the field $\displaystyle F$ has characteristic $\displaystyle 0$, prove that $\displaystyle H=\left\{{me/ne:m\in{\mathbb{Z}}},n\in{\mathbb{N}^*}\right\}\su bset F$ is a subfield of $\displaystyle F$ ($\displaystyle e$ is the unit on $\displaystyle F$). Besides, $\displaystyle H$ is isomorphic to $\displaystyle \mathbb{Q}$.
Regards.
Fernando Revilla