This question began as [Why are procyclic subgroups of Galois groups of number fields free profinite?](https://mathoverflow.net/questions/273088/why-are-procyclic-subgroups-of-galois-groups-of-number-fields-free-profinite), which fizzled out, but which garnered some helpful comments from YCor.


Let $K$ be a field, take $\omega \in \mathrm{Gal}(\bar{\mathbb{Q}}/K)$, and let $L\_{\omega}$ be the fixed field of $\omega$. Then $\Gamma = \mathrm{Gal}(\bar{\mathbb{Q}}/L\_{\omega})$ is a procyclic group, and (using Artin-Schreier) must be isomorphic to $$1, \mathbb{Z}/2\mathbb{Z}, \prod\_{p\in S} \mathbb{Z}\_p\textrm{, or }(\mathbb{Z}/2\mathbb{Z}) \times \prod\_{2 \neq p \in S} \mathbb{Z}\_p$$
for a subset $S$ of primes.


**Question:** Which of these groups can occur? Using the normalized Haar measure on $\mathrm{Gal}(\bar{K}/K)$, can one calculate the probability that an element will generate a given procyclic group?


An element $\omega \in \hat{\mathbb{Z}}$ will generate a subgroup isomorphic to $\hat{\mathbb{Z}}$ with probability 1. So this answers the case for finite fields, and also leads me to believe that $\hat{\mathbb{Z}}$ is likely to occur in general, but I don't know whether any $\hat{\mathbb{Z}}$ occurs in general or whether they would be outnumbered by many $\mathbb{Z}\_p$ that do not extend.


I would be most interested in the field $K$ being $\mathbb{Q}$ or a number field, but also in $\mathbb{Q}^{ab}$ and function fields over $\mathbb{F}\_q$ or $\bar{\mathbb{F}\_q}$. The case of number fields may be easier if we assume totally imaginary, to dispense with the $\mathbb{Z}/2\mathbb{Z}$.


It would be interesting to know whether number fields can be distinguished by these probabilities.



What is the probability of generating a given procyclic subgroup in $\mathrm{Gal}(\bar{K}/K)$?
$\hat{\mathbb Z}$ occurs with probability one.


It is easy to define a homomorphism from the Galois group of a number field to $\hat{\mathbb Z}$. To do this, just define a homomorphism to $\mathbb Z\_p$ for each $p$. It is sufficient to define such homomorphisms for $\mathbb Q$, as the restriction to the Galois group of a number field will be an open subgroup, also isomorphic to $\mathbb Z\_p$. Over $\mathbb Q$, this can be done using the $p$-power cyclotomic tower and the $p$-adic logarithm.


Then by the argument you noted, with probability $1$ the image of your element inside this generates a $\hat{\mathbb Z}$. It is easy to see from this that your element generates $\hat{\mathbb Z}$ in the original Galois group.


