nt.number theory - Decomposition of primes, where the residue field extensions are allowed to be inseparable

I've been dealing with the following situation:

Let $Rsubseteq S$ be an extension of Dedekind rings, where $Quot(R)=:L subseteq E:=Quot(S)$ is a $G$-Galois extension. Let $mathfrak{p}$ be a prime of $R$, and $mathfrak{q}$ a primes of $S$ above $mathfrak{p}$. Let $D_{mathfrak{q}}$ denote the decomposition group, and $I_{mathfrak{q}}$ the inertia group, of $mathfrak{q}$ over $mathfrak{p}$.

However, unlike in the classic case, I allow the residue fields of $mathfrak{p}$ to be infinite, with positive characteristic. So the extension of residue fields may be inseparable.

It seems that the paper I'm reading implicitly assumes:

$|I_{mathfrak{q}}|=e[kappa(mathfrak{q}):kappa(mathfrak{p})]_ i$ (the ramification index times the inseparability degree of the residue extension)

$|D_{mathfrak{q}}|=e[kappa(mathfrak{q}):kappa(mathfrak{p})]$

$|G|=re[kappa(mathfrak{q}):kappa(mathfrak{p})]$ (where $r$ is the number of primes above $mathfrak{p}$)

Is that right? I keep hitting walls when I try to prove it.

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