Exercise 7.1.4 (Normal extensions lemma)

Proof. If we drop the word ‘irreducible’ , Lemma 7.1.2 becomes false.

To give a counterexample, consider the extension $\mathbb{Q}(\sqrt{2}):\mathbb{Q}$, which is normal, since $[\mathbb{Q}(\sqrt{2}):\mathbb{Q}]=2$. The polynomial $(t^2-2)(t^2+1)\in \mathbb{Q}[t]$ has a root in $\mathbb{Q}(\sqrt{2})$, namely $\sqrt{2}$, but $i$ is another root which is not in $\mathbb{Q}(\sqrt{2})$. □