Euclidean Domain in the Ring Q\(\sqrt{-43}\): A Mathematical Insight - Ashara, Precious C.; Obi, Martin C. (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 \(\to\mathbb{Z^+}\) 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[\(\sqrt{-43}\)] is not a 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 our previous work. We proved that the ring Q\(\sqrt{-43}\) fails to have universal side divisors and, thus, fails to be Euclidean domain (ED).