Let $p \equiv 1 \bmod 4$ be a prime, and let $\omega=e^{2 \pi i / p}$. Let $L=\mathbb{Q}(\omega)$.

(a) Show that $[L: \mathbb{Q}]=p-1$.

(b) Calculate $\operatorname{disc}\left(1, \omega, \omega^{2}, \ldots, \omega^{p-2}\right)$. Deduce that $\sqrt{p} \in L$.

(c) Now suppose $p=5$. Prove that $\mathcal{O}_{L}^{\times}=\left\{\pm \omega^{a}\left(\frac{1}{2}+\frac{\sqrt{5}}{2}\right)^{b} \mid a, b \in \mathbb{Z}\right\}$. [You may use any general result without proof, provided that you state it precisely.]