determine its $\pi_K$-adic valuation.

Fix representatives $v_1,\ldots,v_s$ of the (finitely many) residue classes in $\mathcal{O}_K^n/(\pi_K^{R_0+1} \mathcal{O}_K)^{n}$, and for every $i=1,\ldots,s$ let $\omega_i=\omega(v_i)$. Write $[\mathcal{O}_L : \mathcal{O}_K[G]   \omega_i]=\pi_K^{R_i} \mathcal{O}_K$. By the same argument as above, each of the finitely many $R_i$ is effectively computable, and there is no loss of generality in assuming that there is an index $i_0$ for which $\omega_{i_0}=\omega_0$. We claim that $m(L/K)$ is equal to the norm of the ideal $\pi_K^{\min_i R_i}$. To prove this, let $$r_i = \min_{v \equiv v_i \bmod{\pi_K^{R_0+1}}} v_K [\mathcal{O}_L : \mathcal{O}_K[G]   \omega(v)]_{\mathcal{O}_K},$$ where the minimum is taken over all vectors $v \in \mathcal{O}_K^n$ all of whose coordinates are congruent to the respective coordinates of $v_i$ modulo $\pi_K^{R_0+1}$. We now show that either $r_i=R_i$ or $r_i {\geqslant}R_0$. Indeed, it is clear by definition that $r_i {\leqslant}R_i$. So $r_i=R_i$ or $r_i < R_i$, and in this last case we can show that $r_i {\geqslant}R_0$. In fact, if we had $r_i < R_0$ the index would be constant on the coset $v_i +\pi_K^{R_0+1}$, and this is not the case since we are assuming $r_i< R_i$. This proves that for all $i$ we have $$\min\{r_i, R_0\} = \min\{R_i, R_0 \};$$ note in particular that for the 