Let $\zeta_n$ be a primitive $n$-th root of unity with $n > 2$.
How to show (supposedly using Galois theory) that the extension $\mathbb{Q}(\zeta_n)/\mathbb{Q}(\zeta_n+\zeta_n^{-1})$ has degree $2$, meaning : $$[\mathbb{Q}(\zeta_n):\mathbb{Q}(\zeta_n+\zeta_n^{-1})]=2$$
The polynomial $$X^2-(\zeta _n+\zeta_n ^{-1})X+1$$ is the minimal polynomial of $\zeta _n$ on $\mathbb Q(\zeta _n+\zeta _n^{-1})$. It's easy to check that it's an annihilator polynomial. The irreducibility come from that fact that if $\zeta _n+\zeta_n ^{-1}\in \mathbb R$ whereas $\zeta _n\notin \mathbb R$, and thus $$[\mathbb Q(\zeta _n):\mathbb Q(\zeta _n+\zeta _n^{-1})]= 2.$$