example of an extension that is not normal

In this entry, $\sqrt[3]{2}$ indicates the real cube root of $2$.

Consider the extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$. The minimal polynomial for $\sqrt[3]{2}$ over $\mathbb{Q}$ is $x^3-2$. This polynomial factors in $\mathbb{Q}(\sqrt[3]{2})$ as $x^3-2=(x-\sqrt[3]{2})(x^2+x\sqrt[3]{2}+\sqrt[3]{4})$. Let $f(x)=x^2+x\sqrt[3]{2}+\sqrt[3]{4}$. Note that . Thus, $f(x)$ has no real roots. Therefore, $f(x)$ has no roots in $\mathbb{Q}(\sqrt[3]{2})$ since $\mathbb{Q}(\sqrt[3]{2})\subseteq ℝ$. Hence, $x^3-2$ has a root in $\mathbb{Q}(\sqrt[3]{2})$ but does not split in $\mathbb{Q}(\sqrt[3]{2})$. It follows that the extension $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not normal.