Quadratic Reciprocity/ Euler's Criterion

**cathwelch** · May 11th 2009, 01:42 PM

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.

**PaulRS** · May 11th 2009, 02:59 PM

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

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