Then:

1.  the inequality $-1 {\leqslant}t {\leqslant}\frac{e_Kp}{p-1}$ holds;

2.  if $p \mid t$, then $t = \frac{e_Kp}{p - 1}$, the ground field $K$ contains the $p$-th roots of unity, and there exists a uniformiser $\pi_K$ of $K$ such that $L = K(\pi_K^{1/p})$.

\[def:AlmostMaximallyRamified\] Let $L/K$ be a ramified cyclic extension of degree $p$. The extension $L/K$ is called *maximally ramified* if its ramification jump assumes its maximum possible value, namely $t=\frac{e_Kp}{p-1}$, and is called *almost maximally ramified* if $t$ satisfies $$\frac{e_Kp}{p-1}-t\le1 .$$

More generally, a totally ramified cyclic extension $L/K$ of degree $dp^n$, with $(d,p)=1$, is called *almost maximally ramified* if its first ramification jump $t_1$ satisfies $$\frac{e_Kdp}{p-1}-t_1\le 1.$$

Jacobinski [@jac64] defined an extension $L/K$ with Galois group $G$ to be almost maximally ramified if all idempotents $e_H=\frac1{|H|}\sum_{\sigma\in H}\sigma$ belong to the associated order $\mathfrak{A}_{L/K}$, when $H$ ranges over all subgroups of $G$ included between two consecutive ramification groups of the extension. Using basic facts on ramification explained in [@MR554237 Chapters 3 and 4], one can verify that $e_H\in\mathfrak{A}_{L/K}$ if and only if $$\label{eq:RamificationInequality}
 \sum_{i=0}^\infty (|G_i(L/L^H)|-1){\geqslant}e_Lv_p(|H|),$$ where $L^H$ is the field fixed by $H$ and $G_i(L/L^H)$ the $i$-th ramification group of the extension $L/L^H$. One may then show that Jacobinski’s 