Let's consider a Dedekind domain $A$ with field of fractions $K$. Let $L$ be a finite Galois extension of $K$ and $B$ the integral closure of $A$ in $L$. Let $\mathfrak{p}\subset A$ be a non-zero prime ideal and $\mathfrak{P} \subset B$ a prime lying above $\mathfrak{p}$. Let $L'$ be the inertia field of $\mathfrak{P}$ and let $\mathfrak{P}':=\mathfrak{P}\cap L'$. Denote by $k(\mathfrak{P})$, $k(\mathfrak{P}')$, $k(\mathfrak{p})$ the respective residue fields.
In class we proved that $k(\mathfrak{P}')$ is the maximal separable subextension of $k(\mathfrak{P})/k(\mathfrak{p})$ and a step of the proof says that the extension $k(\mathfrak{P})/k(\mathfrak{P'})$ is purely inseparable. But to say this I should know that the characteristic of the residue field is positive. How do I know this? Does it follow from some properties of Dedekind domains? Is it some known fact? In my notes there's nothing about it.
The proposition in a more elaborated form) is stated as Thm. 24 on p. 292 of Commutative Algebra I by Zariski and Samuel (pdf), where he also says that the characteristic of the residue field is $p>0$, but I don't see why.