q is power of odd prime.How to show that $-1$ is square in $\mathbb{F_q}$ if $ q=1(mod4)$ i.e order of field divide 4 and -1 is not square if $ q=3(mod4)$. I know that since q is power of odd prime; order of $\mathbb{F_q}^*$ is $q-1$ and its cyclic group of even order. I also know only element...