following proposition, which is a special case of [@CF Proposition 4 on p.12].

\[prop:disc\] Let $X,Y$ be $\mathcal O_K$-lattices such that $ X \otimes_{\mathcal{O}_K} K = Y \otimes_{\mathcal{O}_K} K=:V$. Let $B$ be a non-degenerate symmetric bilinear form on $V$. Then

1.  $\operatorname{disc}_{B}(Y)=[X:Y]_{\mathcal O_K}^2\operatorname{disc}_{B}(X).$

2.  If $X$ is the free $\mathcal O_K$-module generated by $\{x_i\}_{i=1}^n$ then $$\operatorname{disc}_K(X)=\det\{B(x_i,x_j)\}.$$

Preliminaries on ramification groups
------------------------------------

Let $L/K$ be a Galois extension of $p$-adic fields with Galois group $G$.

\[def:RamificationGroups\] For $i {\geqslant}-1$ we let $$G_i = \{g \in G : (g-1)(\mathcal{O}_L) \subseteq (\pi_L)^{i+1} \}$$ be the $i$-th ramification group of $G$ in the lower numbering.

Clearly, $G_{i+1}\unlhd G_i$ for each $i\ge-1$ and we say that $t$ is a (lower) ramification jump for the extension $L/K$ if $G_t\ne G_{t+1}$. It is well-known that for a fixed Galois extension $L/K$ the group $G_i$ is trivial for $i$ large enough (see [@MR554237 Chapter 4] for details). The extension $L/K$ is called *unramified* (respectively *tamely ramified*, *weakly ramified*) when $G_0$ (respectively $G_1, G_2$) is trivial. The unique ramification jump of a ramified extension of degree $p$ has the following property.

\[prop:InequalitiesOneAndt\] Let $L/K$ be a cyclic ramified extension of degree $p$ and let $t$ be its ramification jump. 