Hopf orders in K[Cp3] over a local field

Abstract

Let K be a field of characteristic p or 0 that is complete with respect to a discrete valuation. Let $C_{p^n}$ denote the cyclic group of order $p^n$, $1\le n\le 3$. In the characteristic p case, we construct a new collection of Hopf orders in $K\left[C_{p^3}\right]$. When the characteristic of K is 0 and K contains a primitive pth root of unity, we show that Hopf orders in $K\left[C_{p^2}\right]$ can be translated from truncated exponential form to Gauss sum form and vice versa. We investigate the analogous translation problem in the $C_{p^3}$ case when the characteristic of K is 0 and K contains a primitive $p^3$rd root of unity.