# 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 $\mathbb{Q}(\sqrt[3]{2})\subseteq \mathbb{R}$. 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 |