This is a 4 part question on my tutorial and if you prove part d) the rest follow, except there's no part d) in the question! I can prove a), b) and c) without using whatever d) is but just thought I'd see if you guys know what this proof is actually proving!

Q. Let p be an odd prime, and a, b be integers not divisible by p. Prove that

a) a \equiv b (mod p) implies that

b) .

c) ; .

(I though perhaps part d) was the second equality in part c) but I don't think the proof matches up.

PROOF.

a), b) and c) follow from d) so we should prove that first!

d) Take a primitive root g, with mod .

Then .

As mod we see that...

mod if is even (i.e if a is a quadratic residue),

while mod if is odd (i.e a non quadratic residue). Hence result.

I was thinking it was probably something like show that if g is a primitive root and k is even then,

. and if k is odd then ... = -1.