Euclidean Domain in the Ring Q\(-43\): A Mathematical Insight - Precious C. Ashara; Martin C. Obi (Mathematics and Computer Science: Research Updates Vol. 2, 2025)
It is invalid that any principal ideal domain (PID) is a Euclidean domain (ED): however, the converse is valid. An entire ring R with unity is said to be Euclidean Domain (ED) if on R, we defined a function N: R \(→Z^+\) which admits proper generalization of the Euclidean division of integers. Every Euclidean domain (ED) is a Principal ideal domain (PID), but those that are principal ideal are not Euclidean. We provide detailed proof that the quadratic algebraic integer ring Q[\(-43\)] is not a Euclidean domain. We proved that the ring of algebraic integer in the quadratic complex field Q[\(-43\)] is a principal ideal domain using the developed inequalities and field norm axioms in our previous work. We proved that the ring Q\(-43\) fails to have universal side divisors and, thus, fails to be Euclidean domain (ED).