## You are here

Homeexample of an extension that is not normal

## Primary tabs

# example of an extension that is not normal

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.

## Mathematics Subject Classification

12F10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: numerical method (implicit) for nonlinear pde by roozbe

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag

Aug 24

new question: Scheduling Algorithm by ncovella

new question: Scheduling Algorithm by ncovella