For week 5: Let p be a prime. Describe U(p^n) and find the order of this group. Using your guess from HW3 in Chapter 3 now predict |U(750)|. For week 6: prove that A_3 inside S_3 is normal. For week 7: Show that |U(st)| = |U(s)| * |U(t)| provided that gcd(s,t)=1. Hint: Suppose (c mod st) is in U(st). Show that (c mod s) is in U(s). That sets up a map from U(st) to U(s). Next show that each (a mod s) gets hit U(t) times as follows. Prove that if (c mod st) maps to (a mod s) then we must have c=a+ks for some k between 0 and t-1, and that we want to find the k's in this range for which a+ks is coprime to t. Now recall that gcd(s,t)=1 so that there is an equation xs+yt=1. Conclude a=axs+ayt, feed that into a+ks and conclude that we want the k's in the given range for which ax+k is coprime to t. Notice that there are |U(t)| such k's. For week 12: Show that Phi_n(x) is reducible if n is not prime. For week 16: Without computing its order the long way, show that x is a generator for the group of units of the field ZZ[x]/(2,x^5+x^3+1), For the "recommended exercise": indicate how one might go about constructing a regular pentagon.