Prove of Quadratic residual

Question

Let $p$ be a prime and $q$ a primitive root modulo $p$. How do I show that q is a quadratic residue modulo $p$ if and only if $a \equiv q^{2k}\pmod p$ for some integer $k$?

Answer 1

If $q$ is a is a quadratic residue modulo $p,$ there exists an integer $b$ such that $a\equiv b^2\pmod p$

Now, as $q$ is a primitive root $\pmod p,$ for each distinct integer $k\in[1,p-1],$ we shall have one of the $p-1$ in-congruent integers $b \pmod p$

So, $a\equiv b^2\equiv q^{2k}\pmod p$

Conversely, if $a\equiv q^{2k}\pmod p$ for some integer $k\in[1,p-1]$ and $q$ is a primitive root $\pmod p,$ there exists some integer $b\equiv q^k\pmod p\implies a\equiv q^{2k}\pmod p\equiv(b)^2$