Let p be an odd prime and let a be an integer with p - a.
(a) prove that a(p−1)/2 is congruent to either 1 or −1 modulo p.
(b) prove that a(p−1)/2 is congruent to 1 modulo p if and only if a is a quadratic residue modulo p. (hint. let g be a primitive root for p and usethefact,3.60,thatgm is a quadratic residue if and only if m is even.)
(c) prove that a(p−1)/2 ≡¡a p¢(mod p). (this holds even if p|a.)
(d) use (c) to prove theorem 3.61(a), that is, prove that