definition is equivalent to the inequalities in Definition \[def:AlmostMaximallyRamified\]. See [@MR513880 $\S$1.2] or [@MR543208 Proposition 1] for further details.

General results
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Let $L/K$ be a Galois extension of $p$-adic fields with group $G$. As observed in the Introduction, the quantity $m(L/K)=\min_{\alpha \in \mathcal{O}_L} [\mathcal{O}_L : \mathcal{O}_K[G]  \alpha]$ is well defined as a minimum, and can be computed on the integers which are normal basis generators. In fact, these are precisely the cases for which $\mathcal{O}_K[G]  \alpha$ is a full $\mathcal{O}_K$-submodule of L.

We are interested in bounding the value of $m(L/K)$ in terms of the standard invariants of the extensions $L/K$ and $L/{\mathbb{Q}}_p$. Before doing this, we will show in Theorem \[thm:mLKEffectivelyComputable\] that in principle the minimal index $m(L/K)$ can be effectively computed for any finite Galois extension $L/K$. In practice, direct computation of $m(L/K)$ with the algorithm we propose is limited to very simple extensions only. For this reason, an *a priori* bound on $m(L/K)$ (such as in Theorem \[thm:IntroGeneralBound\]) can be very useful both for the study of a single extension and for understanding the behaviour of $m(L/K)$ in families.

The minimal index $m(L/K)$ is effectively computable
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\[thm:mLKEffectivelyComputable\] Let $L/K$ be a finite Galois extension of 