Definition (Quadratic residue)

The subgroup $\mathbb{F}_{p}^{*2}$ of $\mathbb{F}_{p}^{*}$ (finite field with $p$ prime) has index $2$ and consists of squares. If $g$ is a primitive root of $\mathbb{F}_{p}^{*}$, then $g^{2}$ is a generator of $\mathbb{F}_{p}^{*2}$. Since $g^{\frac{p-1}{2}}\not\equiv1\pmod{p}$ and $$0\equiv g^{p-1}-1\equiv(g^{\frac{p-1}{2}}-1))(g^{\frac{p-1}{2}}+1)\pmod{p}$$ we see that $$g^{\frac{p-1}{2}}\equiv-1\pmod{p}$$ when $g$ is a primitive root $\pmod{p}$ of $\mathbb{F}_{p}^{*}$. $\mathbb{F}_{p}^{*2}$ is called the subgroup of squares. We define elements of $\mathbb{F}_{p}^{*2}$ as quadratic residues.