Q:

countable set that contains 1 and pi and has polynomial with coefficients in set s.t. all real roots are in set

Deduce that there is a countable set X that contains the real numbers 1 and pi and has the further property that if P is any non-zero polynomial with coefficients in X, then all real roots of P belong to X. 
I have absolutley no idea where to start with this, I have managed the rest of the question but I don't really understand what the question is asking, much less how to do it! Any help is appreciated.
Here is the link to my question (question 5) if you want to read it yourself:
http://www.maths.cam.ac.uk/undergrad/pastpapers/2008/Part_IA/PaperIA_4.pdf

A:

HINT: Recall that adding a single real (algebraic, or transcendental) to $\Bbb Q$ results in a countable field; consider its closure.

