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Let p be an odd prime, and let x = [(p−1)/2]!. Prove that x^2 ≡ (−1)^(p+1)/2...

Let p be an odd prime, and let x = [(p−1)/2]!. Prove that x^2 ≡ (−1)^(p+1)/2 (mod p).

(You will need Wilson’s theorem, (p−1)! ≡−1 (mod p).) This gives another proof that if p ≡ 1 (mod 4), then x^2 ≡ −1 (mod p) has a solution.

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