Let \(k\) be a perfect field of characteristic \(p>2\) and \(K\) an extension of \(F=\mathrm{Frac} W(k)\) contained in some \(F(\mu_{p^r})\). Using crystalline Dieudonn\'e theory, we provide a classification of \(p\)-divisible groups over \(\mathscr{O}_K\) in terms of finite height \((\varphi,\Gamma)\)-modules over \(\mathfrak{S}:=W(k)[[u]]\). Although such a classification is a consequence of (a special case of) the theory of Kisin--Ren, our construction gives an independent proof and allows us to recover the Dieudonn\'e crystal of a \(p\)-divisible group from the Wach module associated to its Tate module by Berger--Breuil or by Kisin--Ren.
