For a prime number $p>2$, set $\zeta=e^{2 \pi i / p}, K=\mathbf{Q}(\zeta)$ and $K^{+}=\mathbf{Q}\left(\zeta+\zeta^{-1}\right)$.

(a) Show that the (normalized) minimal polynomial of $\zeta-1$ over $\mathbf{Q}$ is equal to

$$f(x)=\frac{(x+1)^{p}-1}{x} .$$

(b) Determine the degrees $[K: \mathbf{Q}]$ and $\left[K^{+}: \mathbf{Q}\right]$.

(c) Show that

$$\prod_{j=1}^{p-1}\left(1-\zeta^{j}\right)=p$$

(d) Show that $\operatorname{disc}(f)=(-1)^{\frac{p-1}{2}} p^{p-2}$.

(e) Show that $K$ contains $\mathbf{Q}\left(\sqrt{p^{*}}\right)$, where $p^{*}=(-1)^{\frac{p-1}{2}} p$.

(f) If $j, k \in \mathbf{Z}$ are not divisible by $p$, show that $\frac{1-\zeta^{j}}{1-\zeta^{k}}$ lies in $\mathcal{O}_{K}^{*}$.

(g) Show that the ideal $(p)=p \mathcal{O}_{K}$ is equal to $(1-\zeta)^{p-1}$.