Here's part a. The rest follow in a similar fashion.
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
c) ; .
(I though perhaps part d) was the second equality in part c) but I don't think the proof matches up.
a), b) and c) follow from d) so we should prove that first!
d) Take a primitive root g, with mod .
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.
I can prove a), b) and c) fine. My question was about this mystery part d) that only has a proof but the actual question was not on the tutorial.
Basically, I've been given the proof, can you work out what the question was? My lecturers away so I can't ask him right now.
I only included a,b and c so folk can see the whole question and what part d would imply.