Hi all, messed up using Latex at first. Okay, got two problems on quadratic reciprocity. I've worked up something of a proof for each, but I feel very uncertain. Thought maybe someone could give me some pointers.

Okay, simple one first:

Suppose p is a prime such that , and let a be a QR modulo p.

Show that is a solution to

My proof:

a - QR mod p the Legendre symbol

,

so clearly .

Multiplying both sides by a yields

.

But if ,

we have that .

We know that is congruent to a modulo p, so

is a solution to the congruence.

My trouble is that this seems too easy, and not once did I apply .

Any pointers? Thanks in advance.