• May 11th 2009, 02:42 PM
cathwelch
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, 03:59 PM
PaulRS
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)
$