Euclidean Domain in the Ring Q[\(\sqrt{43}\)]
Asian Journal of Mathematics and Computer Research, Volume 30, Issue 4,
Page 6382
DOI:
10.56557/ajomcor/2023/v30i48408
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 nonEuclidean PID in an understandable manner. We hope to look into the formation of these rings, thus, nonEuclidean geometries where the practical application will be more useful. E.g., Elliptic curves on finite fields.
 Entire ring
 squarefree integer
 noneuclidean principal ideal domain
 quadratic complex field
 algebraic integers
 universal side divisor
 field norm
How to Cite
References
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