Subfields generated by the rationals and a primitive root of unity.

Question

Let $\zeta_n$ be the primitive $n$th root of unity and $p,q$ be two distinct primes. Then

1. $\zeta_{12} \notin \mathbb{Q}(\zeta_{31})$.
2. If $p$ divides $q-1$ then $\zeta_p\in \mathbb{Q}(\zeta_q)$.
3. If there exists a field homomorphism from $\mathbb{Q}(\zeta_p)$ to $\mathbb{Q}(\zeta_q)$ then $p-1$ divides $q-1.$
4. $p-1$ divides $q-1$ then $\mathbb{Q}(\zeta_p) \subseteq \mathbb{Q}(\zeta_q)$.
5. $p-1$ divides $q-1$ iff $\mathbb{Q}(\zeta_p) \subseteq \mathbb{Q}(\zeta_q)$.

According to me the first and the second options are false. The third option is correct by using degrees of field extension. Can someone please give me some hint regrading the fourth and fifth options? I have only done a basic course in field theory and have not studied much about the cyclotomic extensions.