1. ## Quadratic Reciprocity/ Euler's Criterion

Question:
If ab = r (modp), where r is a quadratic residue of the odd prime p, prove that a and b are both quadratic residues of p or both non-residues of p.

2. Remember that $\displaystyle x \equiv y\left( {\bmod .p} \right)$ implies $\displaystyle \left( {\tfrac{x} {p}} \right) = \left( {\tfrac{y} {p}} \right)$ so that $\displaystyle 1=\left( {\tfrac{{r}} {p}} \right)= \left( {\tfrac{{ab}} {p}} \right)= \left( {\tfrac{a} {p}} \right) \cdot \left( {\tfrac{b} {p}} \right)$ now multiple by $\displaystyle \left( {\tfrac{a} {p}} \right)$ and we have: $\displaystyle \left( {\tfrac{a} {p}} \right) = \underbrace {\left( {\tfrac{a} {p}} \right)^2 }_{ = 1} \cdot \left( {\tfrac{b} {p}} \right)$