Consider a finite Galois extension $F/K$, where $F=K(\alpha)$. Let $G=\text{Gal}(F/K)$. We are looking for intermediate fields. Let $H$ a subgroup. A nice idea is to consider the element $\beta_H=\sum_{\sigma\in H}\sigma(\alpha)$ and the intermediate field $K(\beta_H)$. The elements here are fixed by the elements of $H$. When can we hope that this is the field corresponding to $H$? I am pretty sure that this works with cyclotomic extension $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ with $p$ prime, but I don't know how much can I extend the proof to reach more general cases.
I looked for similar statements in books without finding them, and this seems strange to me, since in my opinion this is a pretty useful result.