Let $p$ be a prime and $a,b,c$ be integers. Prove that if $ab, ac, bc, abc$ are all quadratic residues $\mod {p}$ then so are $a, b$ and $c$.

Question

Let $p$ be a prime and $a,b,c$ be integers. Prove that if $ab, ac, bc, abc$ are all quadratic residues $\mod {p}$ then so are $a, b$ and $c$.

I am really struggling with this question and would appreciate it if someone could explain the answer to me in detail. Thank you very much.

Answer 1

Suppose $a$ is not a quadratic residue, since $bc$ is a quadratic residue it follows $abc$ is not, contradiction.

Answer 2

$abc$ is a quadratic residue modulo $p$ means $$\Big(\frac{abc}{p}\Big)=\Big(\frac{a}{p}\Big)\Big(\frac{b}{p}\Big)\Big(\frac{c}{p}\Big)=1.$$ Without loss of generality, what would happen if $a$ is a quadratic non-residue? Then $(a/p)=-1$ and then the product $(abc/p)$ would equal $-1$. Then, we'd conclude from this that $abc$ is a quadratic non-residue which is a contradiction.