If the divisor P is a prime of the form 4q+1 then the number -1 or P-1 is certainly a residue.
The Attempt at a Solution
First the book told me to prove that 1 and -1 are the only two remainders that are their own reciprocals modP.
Next, I'm not really sure what to do though.
I know that there has to be P-1/2 quadratic residues meaning 4q+1-1/2= 2q residues, which is an even number.
I suppose next I would assume that -1 is NOT a quad residue modP so that i could prove by contradiction.
Listing the quad residues modp as x1, x2....xp-1/2
That would mean that the product of -1 and x1..... xp-1/2 would yield the quadratic non-residues modP.
However, I really don't know where to go from here, the book says to match the residues with the reciprocals, but I really don't know what to do..
any help would be great!!!