I have three prime ideals that "belong" to $\mathbb{Z}[\frac{1 + \sqrt{-199}}{2}]$:
$P = (2, \frac{1 + \sqrt{-199}}{2})$ , $Q = (5, \frac{1 + \sqrt{-199}}{2})$, and $S = (7, 3 - \frac{1 + \sqrt{-199}}{2})$.
I have shown that in the class group C, the elements $[P],[Q], [S]$ have order $9, 9, 3$. I have also shown that $[S] = [P]^6$ or $[S] = [P]^3$. Since C is said to be isomorphic to $\mathbb{Z_9}$, I suspect that means $[P] = [Q]$, i.e. there exists $\lambda \in \mathbb{C}$ such that $P = \lambda * Q$, but I am having trouble finding one. Does anyone have any suggestions on how to find a $\lambda$, or on how to use ideals to show $[P] = [Q]$?
Even if you know that the ideal class group is cyclic of order 9, that doesn't necessarily mean that $[P] = [Q]$. The cyclic group of order 9 has more than one element of order 9. All it means is that either $[P]$ or $[Q]$ can be a generator, and the other is a power of that generator, i.e.: $[P] = [Q]^s$ for some integer $s$.