8
$\begingroup$

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) ?

$\endgroup$
14
  • 5
    $\begingroup$ Definition. $\mathbb{N}$ is the free monoid on one generator, namely 1. Corollary. $0\in\mathbb{N}$. $\endgroup$ Feb 3, 2010 at 9:50
  • 1
    $\begingroup$ @Pete: I thought that was more common amongst, say, French texts than UKian ones, but am drawing on a very limited sample... $\endgroup$
    – Yemon Choi
    Feb 3, 2010 at 10:36
  • 1
    $\begingroup$ @Pete: I'm from Spain, and we use (a,b) rather than ]a,b[. Just my two cents. $\endgroup$ Feb 3, 2010 at 14:13
  • 2
    $\begingroup$ @Chandan, my understanding is that 0 tends to be in N in several European countries, but not in America. I have been told to use the notation Z_{\ge 0} to avoid ambiguity. (Personally I agree with you, but for a slightly different reason - the non-negative integers are precisely the set of cardinalities of finite sets.) $\endgroup$ Mar 6, 2010 at 16:20
  • 2
    $\begingroup$ @Chandan: And what part of the $SACRED_BOOK says that one should not define $\mathbb N$ as the free semigroup on one generator? :) $\endgroup$ Mar 6, 2010 at 16:57

2 Answers 2

3
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ Many thanks, dke. Nice to know that the degree-p case was considered by Ore in the 30s, and that the degree-p^2 case was settled only last year. So it appears that the general case is an open problem. $\endgroup$ Mar 7, 2010 at 3:54
  • $\begingroup$ Updated link for the Lbekkouri article (the one in the answer is now broken). $\endgroup$ Mar 21, 2022 at 13:39
1
$\begingroup$

I happen to have come across an early paper of Jean-Marc Fontaine --- apparently his first paper --- in which he treats a special case of the problem: Proposition 3 gives a criterion for the (totally ramified) extension defined by an Eisenstein polynomial of $p$-power degree to be galoisian with a unique ramification break ($>0$).

$\endgroup$
3
  • $\begingroup$ The link in your answer is broken. $\endgroup$
    – KConrad
    Nov 27, 2022 at 1:44
  • 1
    $\begingroup$ numdam.org/item/SDPP_1967-1968__9_1_A6_0 $\endgroup$ Nov 28, 2022 at 2:04
  • 1
    $\begingroup$ I'll put bibliographic information here in case the link does not work for someone: Extensions finies galoisiennes des corps valués complets à valuation discrète Fontaine, Séminaire Delange-Pisot-Poitou. Théorie des nombres, Volume 9 (1967-1968) no. 1, Talk no. 6, 21 pages. $\endgroup$
    – KConrad
    Nov 28, 2022 at 4:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.