Given a system of rational polynomials in some number of variables with at least one real solution, I want to prove that there exists a solution that is a tuple of algebraic numbers. I feel like this should be easy to prove, but I can't determine how to. Could anyone give me any help?
I have spent a while thinking about this problem, and I can't think of any cases where it should be false. However I have absolutely no idea how to begin to show that it is true, and I can't stop thinking about it. Are there any simple properties of the algebraic numbers that would imply this? I don't want someone to prove it for me, I just need someone to point me in the direction of a proof, or show me how to find a counterexample if it is actually false (which would be very surprising and somewhat enlightening). If anyone knows anything about this problem I would be thankful if they could give me a little bit of help.