Let $K$ be a finite extension of $\mathbb{Q}$. Is it possible that the commutator subgroup of the absolute Galois group of $K$ (considered as an abstract group) is a closed subgroup? This property holds for the absolute Galois groups of $p$-adic fields because they are topologically finitely generated.



Commutator subgroup of the absolute Galois group - a closed subgroup
No, the abstract commutator subgroup $[G\_K,G\_K]$ of the absolute Galois group $G\_K$ of a number field $K$ is never closed:


Write $[G,G]$ for the commutator subgroup of $G$ as an abstract group,
and $c(G)$ for the commutator width of $G$,
i.e. the minimal $n$ such that every element in $[G,G]$ is a product of at most $n$ commutators.
From the Baire category theorem one gets that


$(\*)$ the commutator subgroup $[G,G]$ of a profinite group $G$ is closed if and only if $c(G)$ is finite.


Moreover, for a profinite group $G$ one clearly has


$(\*\*)$ $c(G) = \sup\_{N\lhd\_c G} c(G/N)$


i.e. the commutator width is the supremum of the commutator width of the (continuous) finite quotients.


So in order to show that $c(G\_K)=\infty$ (and hence $[G\_K,G\_K]$ is not closed) it would suffice to exhibit a family of finite quotients of $G\_K$ of unbounded commutator width. One such family is the family of finite $p$-groups, for a fixed prime $p$:


(1) By a celebrated theorem of Shafarevich [1], every such group (in fact every finite solvable group) is the Galois group of a finite Galois extension of $K$.


(2) By a result of Roman'kov (Theorem 2 in [2]), there exists a finitely generated pro-$p$ group $F$ whose (abstract) second commutator group $F^{(2)}$ is not closed. By $(\*)$ that means that the commutator subgroup $F'=[F,F]$ has $c(F')=\infty$. Therefore, by $(\*\*)$, the family of finite quotients of $F'$, all of which are $p$-groups, has unbounded commutator width.
EDIT: For a direct and short proof of this see the comment of YCor below.


[1] <https://en.wikipedia.org/wiki/Shafarevich%27s_theorem_on_solvable_Galois_groups>


[2] <https://link.springer.com/content/pdf/10.1007/BF01987820.pdf>


