Precious, C. Ashara and Martin, C. Obi
(2023)
*Euclidean Domain in the Ring Q[-43].*
Asian Journal of Mathematics and Computer Research, 30 (4).
pp. 63-82.

## Abstract

An entire ring R with unity is said to be Euclidean Domain (ED) if on R, we defined a function N: R

+ which admits proper generalization of the Euclidean division of integers. Every Euclidean domain (ED) is a Principal ideal domain (PID), but not all principal ideals are Euclidean. We provide detailed proof that the quadratic algebraic integer ring Q[

] is not Euclidean domain. We proved that the ring of algebraic integer in the quadratic complex field Q[

] is a principal ideal domain using the developed inequalities and field norm axioms in [1]. We proved that the ring Q[

] fails to have universal side divisors, thus, fails to be Euclidean domain (ED). This article extended the result application of [1] proving that ring R of algebraic integer in complex quadratic fields Q[] for M = 43 is non-Euclidean PID in an understandable manner. We hope to look into the formation of these rings, thus, non-Euclidean geometries where the practical application will be more useful. E.g., Elliptic curves on finite fields.

Item Type: | Article |
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Subjects: | Opene Prints > Mathematical Science |

Depositing User: | Managing Editor |

Date Deposited: | 09 Oct 2023 07:24 |

Last Modified: | 09 Oct 2023 07:24 |

URI: | http://geographical.go2journals.com/id/eprint/2666 |