I recently asked the following question (Galois elements determined by action on $n$-th roots of rationals?). Somehow I could not log back into that account, and instead have ended up with a new one! Many thanks for the helpful comments on that question; they confirmed what I had thought was the case but wasn't sure, as well as clarified and corrected it.
My motivation in asking the question was certain comments I had come across to the effect that no element of $Gal(\overline{\mathbb{Q} }/\mathbb{Q})$ other than the identity and complex conjugation could be named, i.e., fully characterized because, I assume, in order to do so one would have to specify its action on all algebraic numbers compatibly with all their algebraic relations.
In lieu of full characterization I wondered if it might be possible and useful to characterize collections of Galois elements by properties of their actions on certain distinguished collections of algebraic numbers, e.g., $\mathcal{R} = \{r^{1/n}: r \in \mathbb{Q}, n \in \mathbb{N} \}$. One could, e.g., ask which elements of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ fix $\mathcal{R}$, and study their properties further.
Is that sensible, possible and useful?
