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Steinberg Cubase 4.5.1 Crack is the most efficient program for working with a DAW. It has the best command line, it is light weight. It works at the same speed in SPEED mode and LIMIT mode. And it has advance features to learn your DAW from you.Q:

Representation of an ideal class in $\widehat{\mathbb{Z}}$

Let $n$ be an integer. Let $I(n)$ be the ideal of $R=\mathbb{Z}[\zeta_8]$ generated by $1-\zeta_8^n$. Let $J(n)$ be the maximal ideal of $R$.
Then
$I(n) = (1-\zeta_8^n)R$ where $R=\mathbb{Z}[\zeta_8]$ and $\mathbb{Z}[\zeta_8]=\{a+b\zeta_8:a,b\in \mathbb{Z}\}$.
Consider the ring $\mathbb{Z}/J(n)\mathbb{Z}$. It is of order $2$ and it is isomorphic to $\mathbb{Z}[\zeta_4]$. In this ring, we can represent the ideal $I(n)$ as $(1-\zeta_4^n)R$.
Question :
Since $\mathbb{Z}/J(n)\mathbb{Z}\simeq \mathbb{Z}[\zeta_4]$ and since $a+b\zeta_4=a+b\zeta_4^3$, can we represent $I(n)$ as $(1-\zeta_4^n)R$ in $\mathbb{Z}/J(n)\mathbb{Z}$?
If the answer is yes, I’d also like to know the proof. If it is no, then is there a way to understand why the answer is no? In particular, the representation of $I(n)$ is important for me.
Edit:
With $\zeta_n=e^{2\pi i/n}$, $I(n)$ is the ideal of $R$ generated by $1-\zeta_8^n$. So \$I