*not* free over $\mathcal{O}_K[G]$. More specifically, we measure the failure of freeness by studying the minimal index of a free $\mathcal{O}_K[G]$-module inside $\mathcal{O}_L$, namely the quantity $$m(L/K) := \min_{\alpha \in \mathcal{O}_L} [\mathcal{O}_L : \mathcal{O}_K[G] \alpha],$$ where $[\mathcal{O}_L : \mathcal{O}_K[G]\alpha]$ denotes the group index. This quantity is well defined since for every integral normal basis generator $\alpha$ of the extension the index $[\mathcal{O}_L : \mathcal{O}_K[G]\alpha]$ is finite. Clearly, by Theorem \[tamepadic\], we have $m(L/K)=1$ if and only if the extension is at most tamely ramified. Notice that, even though this is not clear from the definition, $m(L/K)$ is in principle computable for any finite extension $L/K$: this is the content of Theorem \[thm:mLKEffectivelyComputable\]. In certain settings it may be more natural to consider indices of the form $[\mathcal{O}_L : \mathfrak{A}_{L/K} \alpha]$, but, as we will see, these differ from $[\mathcal{O}_L : \mathcal{O}_K[G] \alpha]$ only by a constant factor independent of $\alpha$ (cf. Proposition \[productindex\]). On the other hand, the proportionality factor $[\mathfrak{A}_{L/K} : \mathcal{O}_K[G]]$ also carries interesting arithmetic content, so that $m(L/K)$ might in fact capture more complete information than the indices $[\mathcal{O}_L : \mathfrak{A}_{L/K} \alpha]$.

It is not the first time that the quantity $m(L/K)$ appears in the literature, in 