The following example is drawn from Milne's Galois Theory notes, p.42 (http://www.jmilne.org/math/CourseNotes/FT.pdf)
We study the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ where $\zeta=e^{2\pi i/7}.$
We find that $\mathbb{Q}[\zeta]$ is the splitting field for the minimal polynomial $x^7-1,$ and that it is a degree 6 Galois extension over $Q.$
We let $\sigma$ be the element of $\text{Gal}(\mathbb{Q}[\zeta]/\mathbb{Q}) = (\mathbb{Z}/7\mathbb{Z})^{\times}$ such that $\sigma \zeta= \zeta^3.$ Note that $\sigma$ is a generator for $\text{Gal}(\mathbb{Q}[\zeta]/\mathbb{Q}).$
We now ask: what is the subfield $S$ of $\mathbb{Q}[\zeta]$ which corresponds to the order 2 subgroup $<\sigma^3>.$
We note that $\sigma^3 \zeta=\zeta^6= \overline{\zeta}.$ So in particular, $\zeta+ \overline{\zeta}$ is fixed by $\sigma^3$ and so $\mathbb{Q}[\zeta+\overline{\zeta}] \subset S.$
Milne claims that we also have $S \subset \mathbb{Q}[\zeta+\overline{\zeta}].$ While this seems reasonable, is there a systematic way to see this?
Hint: $2=[\mathbb{Q}[\zeta] : S]=\left[\mathbb{Q}[\zeta]:\mathbb{Q}[\zeta+\overline{\zeta}]\right]\left[\mathbb{Q}[\zeta+\overline{\zeta}]:S\right]$