Let $p , q$ be distinct prime numbers and let $\zeta _ { p } , \zeta _ { q }$ denote (any) primitive $p$-th and $q$-th roots of unity, respectively. Choose all the correct statements.\\
(A) $\zeta _ { 13 } \notin \mathbb { Q } \left( \zeta _ { 31 } \right)$.\\
(B) If $p$ divides $q - 1$, then $\zeta _ { p } \in \mathbb { Q } \left( \zeta _ { q } \right)$.\\
(C) If $\zeta _ { p } \in \mathbb { Q } \left( \zeta _ { q } \right)$, then $p - 1$ divides $q - 1$.\\
(D) If there exists a field homomorphism $\mathbb { Q } \left( \zeta _ { p } \right) \longrightarrow \mathbb { Q } \left( \zeta _ { q } \right)$, then $p - 1$ divides $q - 1$.