## Euclidean Domain in the Ring Q[\(\sqrt{-43}\)]

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.

*Author to whom correspondence should be addressed.

### 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 non-Euclidean PID in an understandable manner. We hope to look into the formation of these rings, thus, non-Euclidean geometries where the practical application will be more useful. E.g., Elliptic curves on finite fields.

Keywords: Entire ring, square-free integer, non-euclidean principal ideal domain, quadratic complex field, algebraic integers, universal side divisor, field norm

#### How to Cite

*Asian Journal of Mathematics and Computer Research*,

*30*(4), 63–82. https://doi.org/10.56557/ajomcor/2023/v30i48408

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### References

Martins CO, Precious CA. On some non-euclidean principal ideal domain. Asian Journal of Mathematics and Computer Research. 2019;25(8):478-510.

Kuku AO. Abstract Algebra, University of Ibadan. Ibadan University Press Ibadan-Nigeria. ISBN: 978 121 069 9; 1980.

Samuel P. About euclidean rings. Journal of Algebra.1971;19:282–301.

Motzkin T. The euclidean algorithm. Bull. Amer. Math. Soc. 1949;55:1142–1146 4.

Wilson JC. A principal ideal ring that is not a euclidean ring. University of North Carolina at Asheville. Mathematics Association of America: Mathematics Magazine. 1973.46(1):34-48 Zbl284,13018.

Williams KS. Note on non-euclidean principal ideal domain. Math. Mag. 1975;48(3):176-177.

Campoli OA. A principal ideal domain that is not a euclidean domain. American. Mathematics. Monthly 1988;95(9):868-871.

Peric V, Vukovic M. Some examples of principal ideal domain which are not euclidean and some other counterexamples. NOVI Sad Journal of Mathematics.2008;38(1):137-154.

Conan Wong. On a principal ideal domain that is not a euclidean domain. Department of Mathematics, University of British Columbia; 2013.

Available:Conan@math.ubc.ca. www.m-hikari.com>Imf-2013

Wilson RA. An example of a principal idea domain which is not euclidean domain; 2015.

[Robert A. Wilson’s website, accessed on 21st August, 2017].

Available:www.maths.qmul.ac.uk/~raw/MTH5100/PIDNOTED

Precious C Ashara. Determination of some principal ideal domains that are non-euclideans. M.Sc. Thesis. Federal University of Technology, Owerri; 2019.

Hardy GH, Wright EM. An introduction to the Theory of Numbers. 5th Edn. Oxford University Press Walton, Oxford. New York, ISBN 0 19 853171 0; 1979.

Stark HM. A complete determination of the complex quadratic fields of class-number one. Michigan Math. J. 1967;14:1-27.

Michiel Hazewinkel, Nadiya Gubareni, KirichenkoVV. Algebras, rings and modulus. Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow. ISBN 1-4020-2690-0.2004;1.