  Let $K$ be a discretely valued field with ring of integers $\mathcal{O}_K$
with perfect residue field. Let $K(x)$ be the rational function field in one
variable. Let $\mathbb{P}^1_{\mathcal{O}_K}$ be the standard smooth model of
$\mathbb{P}^1_K$ with coordinate $x$ on irreducible special fiber. Let $f(x)
\in \mathcal{O}_K[x]$ be a monic irreducible polynomial with corresponding
divisor of zeroes $\text{div}_0(f)$ on $\mathbb{P}^1_{\mathcal{O}_K}$. We give
an explicit description of the minimal embedded resolution $\mathcal{Y}$ of the
pair $(\mathbb{P}^1_{\mathcal{O}_K}, \text{div}_0(f))$ by using Mac Lane's
theory to write down the discrete valuations on $K(x)$ corresponding to the
irreducible components of the special fiber of $\mathcal{Y}$.

Algebraic Geometry