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
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.