For the sake of simplicity consider $\mathbb{Q}[\zeta_{5}]$. If a cyclotomic integer $z\in\mathbb{Z}[\zeta_{5}]$ is a prime of the integer ring, is it true that its Galois conjugates $\{z, \sigma_1(z), \sigma_2(z), \sigma_3(z)\}$ are primes as well?
If so, is this true more generally?
If $R$ is a commutative unital ring, $p\in R$ is a prime element and $\sigma\in\operatorname{Aut}(R)$, then for all $a,b\in R$ \begin{eqnarray*} \sigma(p)\mid ab \quad&\iff&\quad p\mid\sigma^{-1}(a)\sigma^{-1}(b)\\ \quad&\iff&\quad p\mid\sigma^{-1}(a)\ \vee\ p\mid\sigma^{-1}(b)\\ \quad&\iff&\quad \sigma(p)\mid a\ \vee\ \sigma(p)\mid b, \end{eqnarray*} which shows that $\sigma(p)$ is also prime