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.

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- May 11th 2009, 01:42 PMcathwelchQuadratic 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. - May 11th 2009, 02:59 PMPaulRS
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)

$