Let $p$ be a prime number, $K$ a finite extension of $\mathbb{Q}\_p$, $\mathfrak{o}$ its ring of integers, $\mathfrak{p}$ the unique maximal ideal of $\mathfrak{o}$, $k=\mathfrak{o}/\mathfrak{p}$ the residue field, and $q=\operatorname{Card} k$.


Recall that a polynomial $\varphi=T^n+c\_{n-1}T^{n-1}+\cdots+c\_1T+c\_0$ ($n>0$) in $K[T]$ is said to be **Eisenstein** if $c\_i\in\mathfrak{p}$ for $i\in[0,n[$ and if $c\_0\notin\mathfrak{p}^2$. 


**Question.** When is the extension $L\_\varphi$ defined by $\varphi$ galoisian (resp. abelian, resp. cyclic) over $K$ ?


**Background.** Every Eisenstein polymonial $\varphi$ is irreducible, the extension $L\_\varphi=K[T]/\varphi K[T]$ is totally ramified over $K$, and every root of $\varphi$ in $L\_\varphi$ is a uniformiser of $L\_\varphi$. There is a converse.


If the degree $n$ of $\varphi$ is prime to $p$, then the extension $L\_\varphi|K$ is tamely ramified and can be defined by the polynomial $T^n-\pi$ for some uniformiser $\pi$ of $K$. Thus $L\_\varphi|K$ is galoisian if and only if $n|q-1$, and, when such is the case, $L\_\varphi|K$ is actually cyclic.


**Real question**. Is there a similar criterion, in case $n=p^m$ is a power of $p$, for deciding if $L\_\varphi|K$ is galoisian (resp. abelian, resp. cyclic) ?



When is the extension defined by an Eisenstein polynomial galoisian or abelian or cyclic ?
In the case where the ground field $K$ is $\mathbb{Q}\_p$, some old work of Lbekkouri has recently been published [here](https://doi.org/10.1007/s00013-009-0026-3 "On the construction of normal wildly ramified extensions over Q_p, (p \neq 2). Arch. Math. 93, 331 (2009)"). In particular, for that case, i.e. for finite totally wildly ramified extensions of $\mathbb{Q}\_p$, normality is equivalent to cyclicity. Furthermore:


When $n=p$, this was answered by Ore in the 30's:
the extension is normal if and only if $p^2|c\_j$ for $1\leq j\leq p-2$ and $p^2|(c\_0+c\_{p-1})$.


When $n=p^2$, Lbekkouri gives a list of necessary and sufficent congruence conditions on the coefficients $c\_j$.


More generally for $n=p^m$, he gives some necessary conditions but since the methods require detailed computations with the ramification filtration, it seems unlikely that one could extend the sufficient conditions much beyond the $p^2$ case.


