Field Extensions of Countable Sets

Question

I was wondering, for a field F that is a countable set, and E/F algebraic, is field E necessarily countable as well?

I know algebraic field extension implies every element of E is a root of a polynomial in F[x]. However, I'm not sure how to show if E is countable or not.

Thanks for your help.

Answer 1

So every element of $E$ is in the set of the roots of polynomials over $F$. What is the cardinality of this set? Hint: a countable union of countable sets is still countable.