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This question is from Victor Shoup's book on number theory chapter 2. The problem statement is as mentioned in the title of the question. I haven't been able to crack this one till now. I focused on solving this using the following information:
If p is an odd prime then $(p-1)! = -1 \pmod p$, otherwise called Wilson's theorem.
When $p \equiv 1 \pmod 4$ then any non square in $Z_p^*$ yields a square root of -1 modulo p.
I think we have to prove here that $b=((p-1)/2)!$ doesn't belong to $(Z_p^*)^2$ but I am unable to use Wilson's theorem or any other result to prove it.
PS: Not a homework for me but might be for someone else one day so tagging it as the same.