Suppose I have an $R$-ideal $I$ with $I=(1-\zeta)^n XR$ with $R=\mathbb{Z}[\zeta]$, $\zeta$ a primitive $p$-th root, $X$ an ideal in the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\zeta + \zeta^{-1}]$ (would it be $\mathbb{Z}[\zeta + \zeta^{-1}]$ ?) and the exponent $n$ is $0$ or $1$. Why is such an ideal ambiguous? I can calculate it just for the case where $X$ is ambiguous. But it should work in general.
Thanks for help!
p.s. maybe anyone even knows where I could find the proof for the other direction too? (Each ambiguous $R$-ideal can be written in the form above)