 Solution

Again set $ϵ=(a/p)\in \{1,-1\}$. The Euler criterion applied to $ϵ$ gives $\chi (ϵ)={ϵ}^{(p-1)/2}$, as an equality of signs. If $p\equiv 1(\mathrm{mod}4)$, the exponent is even and the result is $1$; if $p\equiv 3(\mathrm{mod}4)$, it is odd and the result is $ϵ$. Hence