Properties of GF(p) (a) Show that, if p(x) and q(x) are polynomials over the reals (or complex, or rationals) and p(x) q(x) = 0 for all x, then either p(x) = 0 for all x or q(x) = 0 for all x or both. (Hint: You may want to prove first this lemma, true in all fields: The roots of p(x)q(x) is the union of theroots of p(x) and q(x).) (b) Show that the claim in part (a) is false for finite fields GF(p).