# 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 $\operatorname{disc}(f(x))=(\sqrt[3]{2})^{2}-4\sqrt[3]{4}=\sqrt[3]{4}-4\sqrt[3]% {4}=-3\sqrt[3]{4}<0$. Thus, $f(x)$ has no real roots. Therefore, $f(x)$ has no roots in $\mathbb{Q}(\sqrt[3]{2})$ since $\mathbb{Q}(\sqrt[3]{2})\subseteq\mathbb{R}$. 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.

Title example of an extension that is not normal ExampleOfAnExtensionThatIsNotNormal 2013-03-22 16:00:28 2013-03-22 16:00:28 Wkbj79 (1863) Wkbj79 (1863) 6 Wkbj79 (1863) Example msc 12F10