Example 1
(1) Prove: If $a$ is a quadratic residue modulo $p$, then $a \cdot 1^{2}, a \cdot 2^{2}, \cdots, a \cdot\left(\frac{p-1}{2}\right)^{2}$ are exactly all the quadratic residues modulo $p$;
(2) Prove: If $b$ is a quadratic non-residue modulo $p$, then $b \cdot 1^{2}, b \cdot 2^{2}, \cdots, b \cdot\left(\frac{p-1}{2}\right)^{2}$ are exactly all the quadratic non-residues modulo $p$.