$p$-adic fields with Galois group $G$. The quantity $m(L/K)$ may be determined through a finite, effective procedure.

We fix a basis $\alpha_1,\ldots,\alpha_n$ of $\mathcal{O}_L$ over $\mathcal{O}_K$ and use it to identify $L$ with $K^n$ and $\mathcal{O}_L$ with $\mathcal{O}_K^n$: for a vector $v=(c_{1},\ldots,c_{n}) \in \mathcal{O}_K^n$, we let $\omega(v) \in \mathcal{O}_L$ be the element $\sum_{i=1}^n c_i \alpha_i$. Note that an integral basis may be computed effectively, for example using the Montes algorithm [@MR3105943; @MR3276340].

Let $g_1=\operatorname{Id},\ldots,g_n$ be an enumeration of the elements of $G$. Every $g_i$ corresponds to a $K$-linear transformation of $K^n$, which can be represented by a matrix $M_i \in \operatorname{Mat}_{n}(K)$. Notice that $M_k=\{m_{ij}^{(k)}\}$ where $g_k(\alpha_j) = \sum_{i=1}^n m_{ij}^{(k)} \alpha_i$. Since the $\alpha_j$’s are a basis of $\mathcal{O}_L$ over $\mathcal{O}_K$ it follows that the entries of $M_k$ lie in $\mathcal{O}_K$.

Let $\omega_1, \omega_2 \in \mathcal{O}_L$ and let $v_1, v_2$ be the vectors of the coordinates of $\omega_1,\omega_2$ in the basis $\alpha_j$. Assume that $\omega_1$ generates a normal basis and let $[\mathcal{O}_L : \mathcal{O}_K[G]\omega_1]_{\mathcal{O}_K} = \pi_K^{R}\mathcal{O}_K$. With the previous notation, for $i=1,2,$ we have $$\label{indici}
[\mathcal{O}_L : \mathcal{O}_K[G] \omega_i] = N\left( \det \left( v_i \bigm\vert M_2v_i \bigm\vert \cdots \bigm\vert M_n v_i \right) \right).$$

We claim that if $v_2 \equiv v_1 \pmod{\pi^{R+1}}$ 