to the totally ramified case {#subsect:ReductionToTotallyRamifiedCase}
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Let $L/K$ be a Galois extension of $p$-adic fields, $L^{nr}$ be its maximal unramified subextension, and $G_0$ be the inertia subgroup of $G=\operatorname{Gal}(L/K)$. In this section we show that $[\mathfrak{A}_{L/K} : \mathcal{O}_K[G]]$ is bounded above by the analogous quantity for the extension $L/L^{nr}$, and discuss some cases in which the equality $m(L/K)=m(L/L^{nr})$ holds. To compare $\mathfrak{A}_{L/K}$ and $\mathfrak{A}_{L/L^{nr}}$ we start with a result of Jacobinski (see the beginning of $\S2.1$ in [@MR513880]), according to which we have $$\mathfrak{A}_{L/K}=\bigoplus_{s\in G/G_0}(\mathfrak{A}_{L/L^{nr}}\cap K[G_0])s,$$ where $G/G_0$ denotes a fixed system of left coset representatives. Then $$\begin{split}
   [\mathfrak{A}_{L/K}:\mathcal{O}_K[G]]&=[\oplus_{s\in G/G_0}(\mathfrak{A}_{L/L^{nr}}\cap K[G_0])s:\oplus_{s\in G/G_0}\mathcal{O}_K[G_0]s]\\
   &=[\mathfrak{A}_{L/L^{nr}}\cap K[G_0]:\mathcal{O}_K[G_0]]^{[G:G_0]}\\
   &=[\mathcal{O}_{L^{nr}}\otimes_{\mathcal{O}_K}(\mathfrak{A}_{L/L^{nr}}\cap K[G_0]) :\mathcal{O}_{L^{nr}}[G_0]]
  \end{split}$$ (we are using that $\mathcal{O}_{L^{nr}}$ is free over $\mathcal{O}_K$ of rank $[G:G_0]$). As noted for instance by Bergé in [@MR513880], we always have an injection $$\label{injection}
 \mathcal{O}_{L^{nr}}\otimes_{\mathcal{O}_K}(\mathfrak{A}_{L/L^{nr}}\cap K[G_0])\hookrightarrow \mathfrak{A}_{L/L^{nr}}.$$ We have thus obtained the following.

\[indexassociated\] In the above notation, $$[\mathfrak{A}_{L/K}:\mathcal{O}_K[G]]{\leqslant}[\mathfrak{A}_{L/L^{nr}}:\mathcal{O}_{L^{nr}}[G_0]],$$ with equality if and only if (\[injection\]) is a surjection.

However, the injection is not always a surjection. In the same article, Bergé computed the image of when $G_1$ is cyclic, see [@MR513880 Remarque after Lemme 5]. She also gave an elegant description of the image of in the general 