11. Let $p$ be a prime, $\alpha \geqslant 1$. Prove:
(i) For a fixed $p$, all principal characters modulo $p^{\alpha} (\alpha \geqslant 1)$ are the same;
(ii) For each non-principal character modulo $p^{\alpha}$, there exists a unique modulus $k^{*}=p^{\lambda}(1 \leqslant \lambda \leqslant \alpha)$ and a unique primitive character modulo $k^{*}$ that is identical to it;
(iii) For a fixed $\alpha \geqslant 1$, for each character modulo $p^{\lambda}(1 \leqslant \lambda \leqslant \alpha)$, there exists a unique character modulo $p^{\alpha}$ that is identical to it.