Problem 4. The (real) algebraic numbers are defined to be A= {a 2T there is a polynomial P(z) with integer coefficients } of degree at least one such that P(a) = 0. e Prove that Q C A. Prove that vk A for all k N. Prove that Vk A for all k,n N. e Use the previous problems to prove that A is countable. Hint: You can use the fundamental theorem of algebra which says that any degree n polynomial with real coefficients has at most n solutions in R.1