Let $R$ be a complete discrete valuation ring (DVR) of mixed characteristic $(0,p)$ with field of fractions $K$ containing the $p$ th roots of unity. This article is concerned with semistable models of $p$ -cyclic covers of the projective line $C \longrightarrow {\mathbb P}^{1}_{K}$ . We start by providing a new construction of a semistable model of $C$ in the case of an equidistant branch locus. If the cover is given by the Kummer equation $Z^p=f(X_0)$ , we define what we call the monodromy polynomial ${\mathcal L}(Y)$ of $f(X_0)$ , a polynomial with coefficients in $K$ . Its zeros are key to obtaining a semistable model of $C$ . As a corollary, we obtain an upper bound for the minimal extension $K'/K$ , over which a stable model of the curve $C$ exists. Consider the polynomial ${\cal L}(Y)\prod(Y^p-f(y_i))$ , where the $y_i$ range over the zeros of ${\cal L}(Y)$ . We show that the splitting field of this polynomial always contains $K'$ and that, in some instances, the two fields are equal