Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $\alpha \in \overline{\mathbb{Q}}\setminus \mathbb{Q}$ and let $K \subset \overline{\mathbb{Q}}$ be a maximal extension of $\mathbb{Q}$ in respect to not containing $\alpha$ (so $\alpha \notin K$, but $\alpha$ in every nontrivial extension of $K$). Let $G$ be the Galois group of $\overline{Q}$ over $K$. Show that either $G = \mathbb{Z}/2\mathbb{Z}$ or $G = \mathbb{Z}\_p$ ($p$-adic integers) for some prime $p$.