This question began as 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.