# $\u2102$ is not an ordered field

###### Theorem 1.

$\u2102$ is not an ordered field.

First, the following theorem will be proven:

###### Theorem 2.

$\mathbb{Z}[i]$ is not an ordered ring.

###### Proof.

Many facts that are used here are proven in the entry regarding basic facts about ordered rings.

Suppose that $\mathbb{Z}[i]$ is an ordered ring under some total ordering^{} $\le $. Note that $$ and $$

Note also that $i\ne 0$. Thus, either $i>0$ or $$. In either case, $-1=i\cdot i\ge 0\cdot i=0$, a contradiction^{}.

It follows that $\mathbb{Z}[i]$ is not an ordered ring. ∎

Because of theorem 2, no ring containing $\mathbb{Z}[i]$ can be an ordered ring. It follows that $\u2102$ is not an ordered field.

Title | $\u2102$ is not an ordered field |
---|---|

Canonical name | mathbbCIsNotAnOrderedField |

Date of creation | 2013-03-22 16:17:25 |

Last modified on | 2013-03-22 16:17:25 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 10 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 06F25 |

Classification | msc 13J25 |

Classification | msc 12J15 |

Related topic | Complex |

Related topic | BasicFactsAboutOrderedRings |