Exercise 5.2.4

Proof. $\bar{\mathbb{Q}}$ is by definition the set of all complex algebraic numbers over $\mathbb{Q}$. Theorem 4.4.10 shows that $\bar{\mathbb{Q}}$ is an algebraically closed field. If $f\in \mathbb{Q}[x]$ is an irreducible polynomial over $\mathbb{Q}$, a fortiori $f\in \bar{\mathbb{Q}}[x]$, and by definition of an algebraically closed field, $f$ splits completely over $\bar{\mathbb{Q}}$. Thus $\mathbb{Q}\subset \bar{\mathbb{Q}}$ is a normal extension. In Exercise 4.4.1, we showed that $[\bar{\mathbb{Q}}:\mathbb{Q}]=\infty$. This extension is so an example of a normal extension of $\mathbb{Q}$ that is not finite. □