if q be a power of an odd prime. we need to prove that an element alpha belongs to the multiplicative group Fq star, is a square element of Fq star if and only if alpha ^(q-1)/2
=1 . in particular -1 belongs to Fq star square if and only if q is congreunt to 1 (mod4).
(Fq star is the multiplicative group that has size q-1).