It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedean field $K$ has solvable (in fact supersolvable) Galois group $G$. One sees this by using the ramification filtration $G_i=\{g\in G: g\beta\equiv \beta \pmod {\varpi_K^i}\}$.

More nontrivially, local class field theory tells us if $L/K$ is abelian then $G$ must be a quotient of $K^\times$, whose structure as an abelian group is known explicitly.

In any case, this means the inverse Galois problem over $K$ is quite circumscribed. So is it known precisely which finite groups are Galois groups over $K$?

  • $\begingroup$ Has anyone tried to solve embedding problems? $\endgroup$
    – user19475
    Jun 25 '14 at 16:29
  • 2
    $\begingroup$ @YCor What's the point of editing all these old questions? I also saw that you did this with one of my questions. Please don't do it again. $\endgroup$ Feb 26 '20 at 23:22
  • $\begingroup$ @MartinBrandenburg the point is (mainly) to improve tagging. I understand bumping is inconvenient, so I refrain to to too much (without refraining it would be easily 30 a day!). The reason why it came to your questions is that these were good questions with many upvotes, and I treated in priority such questions (in that case, removing the deprecated tag 'abstract-algebra'). $\endgroup$
    – YCor
    Feb 27 '20 at 5:21
  • 3
    $\begingroup$ I'm glad that this question was bumped as I carelessly state in my course notes that such Galois groups are supersolvable. Mistake fixed. $\endgroup$ Feb 27 '20 at 9:29
  • 1
    $\begingroup$ Link to conversation in chat about this discussion unrelated to the question: chat.stackexchange.com/rooms/10243/conversation/… $\endgroup$
    – YCor
    Feb 27 '20 at 18:21

The short answer is (as far as I am aware) no, but there is a lot that is known. Jannsen and Wingberg have given an explicit presentation for $Gal(\overline{K}/K)$ in the case that the residue characteristic is not $2$ (published in Inventiones Math in 1982/1983), and Volker (1984, Crelle) handles the case when $K$ has residue characteristic $2$ and $\sqrt{-1} \in K$. This does not, however, make it trivial to determine which finite groups are quotients of $Gal(\overline{K}/K)$. Some more information can be obtained from Section VII.5 of "Cohomology of Number Fields" by Neukirch, Schmidt and Wingberg. Here's a paraphrase.

If $K$ is a local nonarchimedean field with residue field of characteristic $p$ (and order $q$), let $G = Gal(\overline{K}/K)$, $T$ be the inertia group, and $V$ be the ramification group. Then $G/T \cong \hat{\mathbb{Z}}$, $T/V \cong \prod_{\ell \ne p} \mathbb{Z}_{\ell}$, and $V$ is a free pro-$p$ group of countably infinite rank. Iwasawa showed that $G/V$ is a profinite group with two generators $\sigma$ and $\tau$ so that $\sigma \tau \sigma^{-1} = \tau^{q}$. Also, the maximal pro-$\ell$ quotient of $G$ is known for all $\ell$. For example, if $\mu_{\ell} \not\subseteq K$ and $\ell \ne p$, the maximal pro-$\ell$ quotient is $\mathbb{Z}_{\ell}$ (i.e. for each positive integer $k$, there is a unique Galois extension $L/K$ of degree $\ell^{k}$, namely the unramified one).


Three comments (which I don't have enough reputation to add as comments):

  • The parenthetical claim in the statement of the question is false: Galois groups of local fields need not be supersolvable. For example, $A_4$ is not supersolvable but it is the Galois group of http://www.lmfdb.org/LocalNumberField/ Another example is $\mathrm{GL}_2(3)$, which is the Galois group of http://www.lmfdb.org/LocalNumberField/
  • Even if we restrict to extensions of $\mathbb{Q}_p$, giving a precise answer to this question is likely to be difficult. For example, $C_2^4$ is not the Galois group of any extension of $\mathbb{Q}_p$ (it has too many index-2 subgroups), nor is $\mathrm{SL}_2(3)$ (as proved by Weil in his "Dyadic Exercises" paper), even though $\mathrm{GL}_2(3)$ is.
  • In general one knows that $G=\mathrm{Gal}(L/K)$ must have a cyclic series of the form $$W\unlhd I\unlhd G$$ in which $W$ is a $p$-group that is also normal in $G$ and $I$ has order prime to $p$ (one can add further constraints imposed by the filtration of $W$ and the action of Frobenius). But as the examples above show, these necessary conditions are not sufficient.

One restriction is, e.g. that the groups occuring as a Galois extension of $K/\mathbf{Q}_p$ have to be generated by $\leq n(K)$ elements for some $n(K) \in \mathbf{N}$ depending on $K$. More precisely, one has $n(K) \leq N + 3$ with $N = [K:\mathbf{Q}_p]$ by [Neukirch-Schmidt-Wingberg], Theorem (7.5.14) (for $p > 2$).


Regarding embedding problems: I have found http://link.springer.com/article/10.1007%2Fs10958-009-9588-7 ("The embedding problem with non-Abelian kernel for local fields", Journal of Mathematical Sciences, September 2009, Volume 161, Issue 4, pp 553-557; Zentralblatt Math: https://zbmath.org/?q=an:05660150), which I unfortunately do not have access to.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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