Question about unique subfields in cyclotomic fields.

According to Galois theory, the sub-field structure of $\mathbb{Q}(\omega_p)$ (where $\omega_p = e^{\frac{2\pi i}{p}}$) corresponds with the subgroup structure of $Aut(\mathbb{Q}(\omega_p))$, the automorphism group of $\mathbb{Q}(\omega_p)$.

Clearly, when $\sigma \in Aut(\mathbb{Q}(\omega_p))$, then $\left(\sigma(\omega_p)\right)^p = \sigma(\omega_p^p) = \sigma(1) = 1$, so $\sigma(\omega_p)$ is again a $p-th$ order root of unity, hence we must have $\sigma(\omega_p) = \omega_p^j$ for some $j \in \{1,2,...,p-1\}$.
Also, defining $\sigma_j$ by $\sigma_j(\omega_p) = \omega_p^j$, we see that $\sigma_j\left((\sigma_k(\omega_p)\right) = \sigma_j(\omega_p^k) = \omega_p^{jk}$, implying that $Aut(\mathbb{Q}(\omega_p))$ is isomorphic to $\mathbb{F}_p^*$, so it is cyclic, generated by $\sigma_g$ where $g$ is a generator of $\mathbb{F}_p^*$.

The subgroups of a cyclic group $<\negthickspace \sigma \negthickspace>$ are $<\negthickspace\sigma^e\negthickspace>$ where $e$ divides the group order, $p-1$ in this case. Thus, for example, for $p=13$ we get the following correspondence between subfields of $\mathbb{Q}(\omega_p)$ and subgroups of $\mathbb{F}_p^*$:

Where $p^*=\left(\frac{-1}{p}\right) p$, in the example $p^*= 13$.

This answers (1) and (2).
I leave (3) and (4) for you :-).