prime element $1-\zeta_5$

Question

We now that $\lambda = 1-\zeta_5$ is the unique prime obove $5$ in $Q(\zeta_5)$, I try to simplify, the congruence relation $n \equiv 1 [\lambda^5]$ to be modulo an intger, is that possible ???

Answer 1

HINT:

Check first that the integers $1-\eta_5^k$, $k=1,2,3,4$, are all associate, that is, their pairwise quotients are integers and moreover units. Then check that $\prod_{k=1}^4(1-\eta_5^k) = 5$.