Euclidean Domain in the Ring Q\(\sqrt{-43}\): A Mathematical Insight
DOI:
https://doi.org/10.9734/bpi/mcsru/v2/3518Keywords:
Entire ring, square-free integer, non-euclidean principal ideal domain, quadratic complex field, algebraic integers, universal side divisor, field normAbstract
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).
