Quad Residues

**cathwelch** · May 13th 2009, 08:51 PM

Question:

Prove: If p = 1(mod 4) is a prime, then -4 and (p-1)/4 are both quadratic residues of p.

**PaulRS** · May 14th 2009, 04:25 AM

Since 4 is a perfect square: $\left( {\tfrac{{ - 4}} {p}} \right) = \left( {\tfrac{{ - 1}} {p}} \right)\left( {\tfrac{4} {p}} \right) = \left( {\tfrac{{ - 1}} {p}} \right) = {\left( { - 1} \right)^{\tfrac{{p - 1}} {2}}} = 1$

Let's denote $a=\left( {\tfrac{{p - 1}} {4}} \right)$ . We have $\left( {\tfrac{a} {p}} \right) = \left( {\tfrac{a} {p}} \right)\left( {\tfrac{{ - 4}} {p}} \right) = \left( {\tfrac{{ - 4a}} {p}} \right) = \left( {\tfrac{{ - \left( {p - 1} \right)}} {p}} \right) = \left( {\tfrac{1} {p}} \right) = 1$

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