Okay, first question on this site, apologies in advance for any mistakes I may make.
Question: So I need to show that an algebraic field extension $E:F$, with $F$ being countable, is countable.
My Thoughts: As far as I am aware, $E$ being algebraic means that every element in $E$ is algebraic (root of some polynomial in $F[x]$ ) and being countable means that there is a natural mapping of the integers onto $E$.
At the moment I am thinking that as every element in $E$ is algebraic then it has some corresponding polynomial in $F[x]$, that it is a root of, and then because $F$ is countable I need a bijection (or possibly just a surjection) of $F$ onto $E$ and that that may be achievable through the connection between an element of $E$ and it's polynomial in $F[x]$. I just can't see how to make this rigorous.
Note: I did try to find something resembling this question but mostly found analysis based questions. This may be down to my unfamiliarity with the website though.