Corollary

If the order of the inertia group $\mathscr{G}^T(p')$ is relatively prime to the characteristic exponent p of k, the field $k'$ is a Galois extension of k.

With the notation of the proof of Proposition 5, the polynomial $\pi'(P)$ has coefficients in $k^T = k'_s$ and all its roots equal to $\pi'(x)$; we immediately deduce that $\pi'(P)$ is a power of a minimal polynomial of $\pi'(x)$ over $k'_s$; but the latter has degree equal to a power of p and hence, as the degree of $\pi'(P)$ is equal to the order of $\mathscr{G}^T$, the hypothesis implies that $\pi'(x) \in k'_s$, in other words $k'_s = k'$.