Quartic residue ($a\in ({\mathbb{F}}_q^{\times }{)}^4$)

A nonzero element $a$ of a finite field is a quartic residue when $a=b^4$ for some nonzero $b$. If $q\equiv 1(\mathrm{mod}4)$, the multiplicative group of a finite field is cyclic of order divisible by four, so the quartic residues form a subgroup of index four; equivalently $a^{(q-1)/4}=1$.