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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

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    $\begingroup$ What is it you want to calculate about these quotients? For comparison, how would you "calculate" ${\mathbf Z}/p^r{\mathbf Z}$ other than saying it is what it is? $\endgroup$
    – KConrad
    May 4, 2014 at 17:22
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    $\begingroup$ Whatever you have in mind, probably it would be helpful to write the quotient as a quotient of a $p$-adic integer ring. The minimal polynomials of $\zeta = \zeta_{p^r}$ over $\mathbf Q$ and $\mathbf Q_p$ are the same, and $\pi := 1 - \zeta$ is a uniformizer of $\mathbf Q_p(\zeta)$, so ${\mathbf Z}[\zeta]/(1 - \zeta)^i \cong {\mathbf Z}_p[\pi]/(\pi)^i$. On the right side you might be able to use the $p$-adic exponential (or logarithm, if you want to focus on the units) once it's clearer what you want to really calculate. $\endgroup$
    – KConrad
    May 4, 2014 at 17:27
  • $\begingroup$ I'm especially interested in the abelian group structure of the quotients in terms of concrete generators (represented by elements in $\mathbb{Z}[\zeta]$) and their relations. The multiplicative structure doesn't matter much to me for the moment as this might be irrelevant in the application I have in mind. $\endgroup$
    – Felix Springer
    May 5, 2014 at 6:55

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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

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