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) ?
Best Answer
In the case where the ground field $K$ is $\mathbb{Q}_p$, some old work of Lbekkouri has recently been published here. 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.