Let $l=p^r$ a prime power and $\zeta$ a primitive l-th root of unity. It is classical result, that $(1-\zeta)^{\varphi(l)}=p\cdot\epsilon\in\mathbb{Z}[\zeta]$ for a unit $\epsilon$.
It should be a classical problem to calculate the quotients $\mathbb{Z}[\zeta]/(1-\zeta)^i$ for nonnegative integers $i$.
Does anyone know a reference for the calculation of these quotients? 


Thanks Felix



Quotients of number rings IZ[zeta_l]
At least to me, KConrad's advice somehow makes the situation a bit clearer and as I was only interested in the structure as an Abelian group everything is pretty easy. $\mathbb{Z}[\zeta]$ is the free Abelian group on the generators $(1-\zeta)^k$ for $k=0,...,p^r-p^{r-1}-1$. As a subgroup the ideal $(1-\zeta)^i$ is generated $(1-\zeta)^{i+k}$ for $k=0,...,p^r-p^{r-1}-1$. As $(1-\zeta)^{\varphi(l)}=p\cdot\epsilon$ for a unit $\epsilon$ we can substitute $(1-\zeta)^j$ by $p^{\lfloor\frac{j}{p^r-p^{r-1}}\rfloor}(1-\zeta)^{j\mod \varphi(l)}$ using the ring structure.
Now the quotient is just the sum of the obvious "partial" quotients.
The multiplicative structure should come easily from $\mathbb{Z}[\zeta]$ though I haven't tried to write it down more explicitly as "saying it is what it is" as KConrad phrased it.


Thanks Felix


