nt.number theory - When is the extension defined by an Eisenstein polynomial galoisian or abelian or cyclic ?

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 $c_iinmathfrak{p}$ for $iin[0,n[$ and if $c_0notinmathfrak{p}^2$.

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) ?

• Next ag.algebraic geometry - Explicit equations for Schubert varieties
• Previous Sets that can be mapped onto R^n by a polynomial