A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163

Victor J. Ramírez V.

doi:10.15446/recolma.v50n2.62206

Published

2016-07-01

A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163

DOI:

https://doi.org/10.15446/recolma.v50n2.62206

Keywords:

Unique factorization domain, prime, irreducible (en)

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Authors

Victor J. Ramírez V. Universidad Simon Bolivar

Abstract (en)

In this paper, we give an elementary proof of the fact that the rings are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedekind and Hasse theorems. Furthermore, it does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings.

DOI: https://doi.org/10.15446/recolma.v50n2.62206

A new proof of the Unique Factorization of for d = 3, 7, 11, 19, 43, 67, 163

Una nueva demostración de la factorización única para d = 3, 7, 11, 19, 43, 67, 163

Victor J. Ramirez V.¹

¹ Universidad Simon Bolivar, Caracas, Venezuela. ramirezv@usb.ve

Abstract

In this paper, we give an elementary proof of the fact that the rings are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedekind and Hasse theorems. Furthermore, it does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings.

Keywords: Unique factorization domain, prime, irreducible.

Mathematics Subject Classification: 11R29, 13F15.

Resumen

En este artículo, damos una demostración elemental de que los anillos son dominios de factorización única para los valores d = 3, 7, 11, 19, 43, 67, 163. Si bien este resultado es conocido, nuestra prueba es nueva y completamente elemental, y no hace uso del teorema del cuerpo convexo de Minkowski, ni del teorema de Dedekind y Hasse. Además, no utiliza la teoría de los enteros algebraicos, ni la teoría de los anillos noetherianos. Sólo utiliza nociones básicas de la teoría de los anillos conmutativos.

Palabras claves: Dominio de factorización única, primo, irreducible.

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References

[1] S. Alaca and K. S. Williams, Introductory algebraic number theory, Cambridge University Press, 2004.

[2] H. Cohn, Advanced number theory, Dover, New York, 1980.

[3] A. Oneto and V. Ramirez, Dominios principales no euclidianos, Divul.Mat. 1 (1993), 55-65.

[4] H. Pollard, The theory of algebraic numbers, Carus Monograph 9, MAA, Wiley, New York, 1975.

Recibido: enero de 2016 Aceptado: marzo de 2016

How to Cite

APA

Ramírez V., V. J. (2016). A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163. Revista Colombiana de Matemáticas, 50(2), 139–143. https://doi.org/10.15446/recolma.v50n2.62206

ACM

[1]

Ramírez V., V.J. 2016. A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163. Revista Colombiana de Matemáticas. 50, 2 (Jul. 2016), 139–143. DOI:https://doi.org/10.15446/recolma.v50n2.62206.

ACS

(1)

Ramírez V., V. J. A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163. rev.colomb.mat 2016, 50, 139-143.

ABNT

RAMÍREZ V., V. J. A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163. Revista Colombiana de Matemáticas, [S. l.], v. 50, n. 2, p. 139–143, 2016. DOI: 10.15446/recolma.v50n2.62206. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/62206. Acesso em: 22 jan. 2025.

Chicago

Ramírez V., Victor J. 2016. “A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163”. Revista Colombiana De Matemáticas 50 (2):139-43. https://doi.org/10.15446/recolma.v50n2.62206.

Harvard

Ramírez V., V. J. (2016) “A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163”, Revista Colombiana de Matemáticas, 50(2), pp. 139–143. doi: 10.15446/recolma.v50n2.62206.

IEEE

[1]

V. J. Ramírez V., “A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163”, rev.colomb.mat, vol. 50, no. 2, pp. 139–143, Jul. 2016.

MLA

Ramírez V., V. J. “A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163”. Revista Colombiana de Matemáticas, vol. 50, no. 2, July 2016, pp. 139-43, doi:10.15446/recolma.v50n2.62206.

Turabian

Ramírez V., Victor J. “A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163”. Revista Colombiana de Matemáticas 50, no. 2 (July 1, 2016): 139–143. Accessed January 22, 2025. https://revistas.unal.edu.co/index.php/recolma/article/view/62206.

Vancouver

1.

Ramírez V. VJ. A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163. rev.colomb.mat [Internet]. 2016 Jul. 1 [cited 2025 Jan. 22];50(2):139-43. Available from: https://revistas.unal.edu.co/index.php/recolma/article/view/62206

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CrossRef Cited-by

4

1. Víctor Julio RAMÍREZ VIÑAS. (2023). NECESSARY AND SUFFICIENT CONDITIONS FOR UNIQUE FACTORIZATION IN ℤ[(-1 + √d)/2]. Kyushu Journal of Mathematics, 77(1), p.121. https://doi.org/10.2206/kyushujm.77.121.

2. Víctor J Ramírez. (2023). Prime producing quadratic polynomials associated with real quadratic fields of class-number one. Periodica Mathematica Hungarica, 86(2), p.495. https://doi.org/10.1007/s10998-022-00487-1.

3. Víctor Julio Ramírez Viñas. (2019). A simple criterion for the class number of a quadratic number field to be one. International Journal of Number Theory, 15(09), p.1857. https://doi.org/10.1142/S1793042119501033.

4. Paul Pollack, Noah Snyder. (2021). A Quick Route to Unique Factorization in Quadratic Orders. The American Mathematical Monthly, 128(6), p.554. https://doi.org/10.1080/00029890.2021.1898875.