Euclidean Domain in the Ring Q[\(\sqrt{-43}\)]


Published: 2023-10-06

DOI: 10.56557/ajomcor/2023/v30i48408

Page: 63-82

Precious C. Ashara *

Department of Mathematics, Federal University of Technology, Owerri, Imo State, Nigeria.

Martin C. Obi

Department of Mathematics, Federal University of Technology, Owerri, Imo State, Nigeria.

*Author to whom correspondence should be addressed.


An entire ring R with unity is said to be Euclidean Domain (ED) if on R, we defined a function N: \(\to\) \(\mathbb{Z}\)+ 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[\(\sqrt{-43}\)] is not Euclidean domain. We proved that the ring of algebraic integer in the quadratic complex field Q[\(\sqrt{-43}\)] is a principal ideal domain using the developed inequalities and field norm axioms in [1]. We proved that the ring Q[\(\sqrt{-43}\)] fails to have universal side divisors, thus, fails to be Euclidean domain (ED). This article extended the result application of [1] proving that ring of algebraic integer in complex quadratic fields Q[\(\sqrt{-M}\)] 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.

Keywords: Entire ring, square-free integer, non-euclidean principal ideal domain, quadratic complex field, algebraic integers, universal side divisor, field norm

How to Cite

Ashara , P. C., & C. Obi , M. (2023). Euclidean Domain in the Ring Q[\(\sqrt{-43}\)]. Asian Journal of Mathematics and Computer Research, 30(4), 63–82.


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