Euclidean Domain in the Ring Q[\(\sqrt{-43}\)]
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.
Abstract
An entire ring R with unity is said to be Euclidean Domain (ED) if on R, we defined a function N: R \(\to\) \(\mathbb{Z}\)+ 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 R 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