is free over $\mathfrak{A}_{L/K}$.

2.  Suppose that the inequality $t < \frac{ep}{p-1} - 1$ holds. If $\mathcal{O}_L$ is free over $\mathfrak{A}_{L/K}$, then $a \mid p-1$.

The inequality $t < \frac{ep}{p-1} - 1$ corresponds to the condition that $L/K$ is not *almost maximally ramified*, see Definition \[def:AlmostMaximallyRamified\].

Finally, a natural related question is to understand which elements generate $\mathcal{O}_K[G]$-submodules of minimal index. We obtain some partial results on this problem by exhibiting an explicit minimal element for cyclic extensions of degree $p$ (Proposition \[prop:aeq0\] and Remark \[rmk:MinimalElement\]).

Acknowledgements {#acknowledgements .unnumbered}
----------------

The authors are indebted to Henri Johnston for his many insightful comments, especially on Theorem \[thm:IntroGeneralBound\], and to Nigel Byott for helpful discussions and for pointing out some relevant results in the literature. We are also grateful to Cornelius Greither for his comments on a preliminary version of the paper and for suggesting Remark \[rmk:greither\].

Preliminaries {#sect:Preliminaries}
=============

Preliminaries on discriminants {#sub:discriminants}
------------------------------

We will use the notation fixed in the Introduction, so $L/K$ is a Galois extension of $p$-adic fields with Galois group $G$. The normal basis theorem ensures that $L/K$ admits a normal basis, namely a $K$-basis of the form $\{\sigma(\alpha)\}_{\sigma\in G}$ for some $\alpha\in L$ (equivalently, we 