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).
Author (s) Details
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.
Please see the book here:- https://doi.org/10.9734/bpi/mcsru/v2/3518
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