Bushnell and Kutzko gave a complete and effective classification of the smooth dual of $\mbox{GL}(N,F)$, where $F$ is a non-archimedean local field. Similarly, Zink gave a classification of the smooth dual of $D^{\times}$, where $D$ is a division algebra with centre $F$, in terms of non-canonical objects and under the restrictive hypothesis that $F$ has characteristic $0$. In this paper, we extend part of Bushnell and Kutzko's formalism to $D^{\times}$ and obtain a complete classification of the smooth dual working for any characteristic. The crucial point of this work is to define a good way of splitting the algebra $D$ so that the important notion of {\it simple stratum}, and its properties, can be translated to $D^{\times}$ by some descent arguments.

1991 Mathematics Subject Classification: 12E15, 20G05, 20G25, 22E50.