Legendre Symbol

**cathwelch** · May 11th 2009, 01:44 PM

Question:
If p is an odd prime and gcd (ab, p)=1, prove that at least one of a, b, or ab is a quadratic residue of p.

**PaulRS** · May 11th 2009, 02:56 PM

Prove it by contradiction, assume none of them were a quadratic residue, then $ \left( {\tfrac{a} {p}} \right) = \left( {\tfrac{b} {p}} \right) = \left( {\tfrac{{ab}} {p}} \right) = - 1 $ however remember that $ \left( {\tfrac{a} {p}} \right) \cdot \left( {\tfrac{b} {p}} \right) = \left( {\tfrac{{ab}} {p}} \right) $

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