is abelian, we establish some relations between the $p$-adic valuation of the minimal index for $E/\mathbb{Q}$, and the $p$-adic valuation of the minimal index for their $p$-adic completions.

Section \[sec:degp\] is devoted to the special interesting case of cyclic extensions of degree $p$. We prove the following exact formula for $m(L/K)$, which holds without any assumption on $K$.

\[thm:IntroCyclicDegreep\] Let $L/K$ be a ramified Galois extension of $p$-adic fields of degree $p$, with ramification jump $t$. Let $a \in \{0,\ldots,p-1\}$ be the residue class of $t$ modulo $p$ and set $\nu_i = \left\lfloor \frac{a+it}{p} \right\rfloor$. Then if $a \neq 0$ we have $
v_p( m(L/K))= f_K \left(\sum_{i=0}^{p-1} \nu_i+\min_{0{\leqslant}i{\leqslant}p-1}(i e_K-(p-1)\nu_i) \right),
$ while for $a=0$ we have $v_p(m(L/K))=\frac{1}{2}[L:\mathbb{Q}_p]$.

The method used to show Theorem \[thm:IntroCyclicDegreep\] also allows us to give, in Section \[sect:BertrandiasFerton\], a new proof of a result originally due to Ferton and Bertrandias [@MR296047 Théorème].

\[thm:IntroFertonBertrandias\] Let $L/K$ be a totally ramified cyclic extension of degree $p$ of $p$-adic fields. Let $t$ be the unique ramification jump of the extension (in the lower numbering) and let $a \in \{0,\ldots,p-1\}$ be the residue class of $t$ modulo $p$. The following hold:

1.  if $a=0$ or $a \mid p-1$, then $\mathcal{O}_L$ 