Proposition 3

Let $K$ be a (not necessarily commutative) locally compact field which is not discrete and whose topology is defined by a discrete valuation $v$; let $A$ be the ring and $m$ the ideal of $v$ and let us write $\mathrm{Card}(A/m) = q = p^f$ (p prime). Then, there exists a system of representatives $S$ of $A/m$ in $A$ and a uniformizer $u$ for $v$ such that $O \in S$, $S^* = S \cap K^*$ is a cyclic subgroup of $K^*$ and $u^{-1} Su = S$. Moreover, every element of $A$ may be written uniquely in the form $\sum_{i=0}^\infty s_i u^i$, where $s_i \in S$.

We shall use the following lemma: