Ghiloni, Riccardo (2009) *The classification of algebraically closed alternative division rings of finite central dimension.* UNSPECIFIED. (Unpublished)

## Abstract

A classical result of Noncommutative Algebra due to I. Niven, N. Jacobson and R. Baer asserts that an associative noncommutative division ring D has finite dimension over its center R and is algebraically closed (that is, every nonconstant polynomial in one indeterminate with left, or right, coefficients in D has a root in D) if and only if R is a real closed field and D is isomorphic to the ring of quaternions over R. In this paper, we extend this classification result to the nonassociative alternative case: the preceding assertion remains valid by replacing the quaternions with the octonions. As a consequence, we infer that a field k of characteristic not 2 is real closed if and only if the ring of octonions over k is an algebraically closed division ring.

Item Type: | Departmental Technical Report |

Department or Research center: | Mathematics |

Subjects: | Q Science > QA Mathematics > QA152 Algebra |

Uncontrolled Keywords: | Algebraically closed rings, Alternative division rings, Real closed fields, Quaternions, Octonions |

Report Number: | UTM 729, June 2009 |

Repository staff approval on: | 23 Jun 2009 |
---|

### Actions (login required)