1) Let p be an odd prime. Prove that there is a primitive root of p

whose (p-1)th power is NOT congruent to 1 mod p^2.

Hint: Let g be any primitive root of p. If g works, done.

If g doesn't work, then show that the primitive root g+p of p

does work.

2) For n > 2, let M = 2^n. Prove by induction on n that

5^(M/8) is congruent to 1 + M/2 mod M.

3) Show that the solutions to x^e - 1 = 0 mod p in RRS(p)

are the d numbers b, b^2, ..., b^d mod p, where d=gcd(e,p-1)

and b is an element mod p of order d.

Hint: Write d as a linear combination...