Problem 2. A real number a is called algebraic, if there exists a polynomial p(r) with integer coefficients such that p(a) = 0. (a) For each n E N, prove that the collection of all polynomials of one variable of degree less than or equal to n with integer coefficients is countable. (b) Prove that the set of all polynomials of one variable with integer coefficients is countable. (c) Prove that the set of all algebraic numbers is countable. (d) A real number that is not algebraic is said to be transcendental. What can you say about the existence of other transcendental numbers? (Remark: It is known that e and * are transcendental. However, the proof of this is far from trivial. In fact, we haven't developed yet the tools to even define the numbers e and 7.)