# quadratic non residue

Printable View

• November 12th 2009, 10:28 PM
scubasteve123
quadratic non residue
Show that the smallest quadratic non residue of an odd prime p is itself prime. Hint: Assume to the contrary and use Legendre Symbols
• November 13th 2009, 09:36 AM
chiph588@
Assume not so let $n$ be the smallest QNR such that $n=ab$. Then $-1=\left(\frac{n}{p}\right)=\left(\frac{ab}{p}\righ t) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right) = (1)(1) = 1$ since $a,b < n$. Hence a contradiction.