Quartic residue
= Quartic residue
{title2=$a\in(\mathbb F_q^\times)^4$}
A nonzero element $a$ of a <finite field> is a quartic residue when $a=b^4$ for some nonzero $b$. If $q\equiv1\pmod4$, 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 $a^{(q-1)/4}=1$.