I have some questions regarding factoring a prime in a cyclotomic field. Say I want to factor 241 in $\mathbb{Z}[\zeta_3]$, I know that I am supposed to look for roots of $x^2+x+1$ in $\mathbb{Z}_{241}$ and that 15 is a root. So now I have $(x-15)$ divides $x^2+x+1$ and how do I proceed from here? According to some examples I have seen, I think I should be looking at the ideals $(\zeta_3-15, 241)$ and $(241)$, and maybe somehow deduce that $(\zeta_3-15, 241)$ is principally generated by $16+15\zeta_3$. My question is 1) how to deduce such an ideal is principally generated and find the value $16+15\zeta_3$ if I do not know I need to use $16+15\zeta_3$ in advance? 2) Why do we need to look at the factorization of the ideal $(p)$ when we are trying to factor p? Is the factorization of the ideal "identical" to the factorization of the prime itself?
Thanks in advance!