Problem. Let p be a prime. Show that the polynomial def x-1+ x-2+ ... +x+1 is ireducible in Q [x]. (Hint: First, note that it is enough to show that Pp (x + 1) is irreducible in Q [x]. Then note that Pp(2 + 1) = (2+1) -1 xtap_12P-1+ ...+ a2x- + px (ac + 1) - 1 = ap-+ ap-12 P-2 + ... + a2x + P, where the fractions are computed in Q(x). We still have the last expression in Z [x], so we have Pp(x + 1) = XP-1+ ap-12P-2+ ...+ a2x + p in Z [x]. Now, show that p divides all a2, a3, . . ., ap-2, ap-1 and apply Eisenstein's criterion.)