-1 as a Square Modulo p

Ponder...

For odd prime : is a square modulo if and only if .

Proposition 4.37

Let be an odd prime. Then is a square modulo if and only if .

Proof

Suppose first that . By Wilson’s theorem,

Since , we have for some , so is even and . Hence

so is a square modulo .

Conversely, we prove the contrapositive: suppose , say . If there were with , then by Fermat’s little theorem,

a contradiction.

When , Wilson’s theorem produces an explicit solution to , namely

since the computation above shows .