a power of the element $1−\zeta_5$

Question

let $\zeta_5$ be the $5-$root of unit and $\lambda = 1-\zeta_5 $ I need to simplify this congruence $n\equiv 1 \pmod{\lambda^5}$ to be modulo an integer in $\Bbb Z$, is that possible?

Answer 1

The ideal generated by $1-\zeta_p$ in $\Bbb Z[\zeta_p]$ ($p$ prime) satisfies $(1-\zeta)^{p-1}=(p)$. Therefore for $p=5$ $$(5)\supset (1-\zeta_5)^5=(1-\zeta)(5)\supset (25)$$ with both containments strict. Therefore, for $n\in\Bbb Z$, $n\equiv1\pmod{(1-\zeta_5)^5}$ iff $n\equiv1\pmod{25}$.