Using Wilson's theorem, show that if $p$ is prime of the form $p=4n+1$ and if $y=(\frac {p-1}2)!$, then $y^2\equiv -1\mod p$.
Show for prime numbers of the form $p=4n+1$,  $x=(2n)!$ solves the congruence $x^2\equiv-1 \pmod p$. $p$ is therefore not a gaussian prime.