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
Canonical name	ExampleOfAnExtensionThatIsNotNormal
Date of creation	2013-03-22 16:00:28
Last modified on	2013-03-22 16:00:28
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	6
Author	Wkbj79 (1863)
Entry type	Example
Classification	msc 12F10