more or less explicit form. For example, in [@MR3411126] Johnston computed an explicit free generator of the ring of integers over its associated order in any wildly and weakly ramified Galois extension $L/K$ of local fields (for some earlier results see also Burns [@MR1760494]); a crucial ingredient in his approach is precisely the computation of $m(L/K)$ in this situation.

\[thm:JohnstonWeaklyRamified\] Let $L/K$ be a wildly and weakly ramified Galois extension of $p$-adic fields. Then $
  m(L/K)=p^{f_L}.
$

In a closely related direction, Köck [@MR2089083] and Johnston [@MR3411126] discuss whether a power of the maximal ideal $(\pi_L)$ can be $\mathcal{O}_K[G]$-free, thus giving some information on the possible indices $[\mathcal{O}_L : \mathcal{O}_K[G]\alpha]$; see also Ullom [@MR263790]. Besides being useful for understanding the *additive* structure of $\mathcal{O}_L$, free $\mathcal{O}_K[G]$-submodules $M$ of the ring of integers also appear in the cohomological study of local class field theory. In fact, starting from such an $M$, one can construct a cohomologically trivial submodule $V$ of finite index in $\mathcal{O}_L^\times$ (see for example [@MR2467155 p. 8]); the cohomology of $\mathcal{O}_L^\times$ is then isomorphic to that of the finite quotient $\mathcal{O}_L^\times/V$, so that information on $[\mathcal{O}_L:M]$ (hence on $[\mathcal{O}_L^\times : V]$) translates into bounds for the cohomology groups 