Python programming and algebra: Some special elements in Gaussian integers modulo a prime


Authors

DOI:

https://doi.org/10.22517/23447214.25748

Keywords:

2-nilpotent; idempotent; invertible; Python; zero-divisor

Abstract

We consider the complexification of a commutative ring with unity and specialize this construction to Z_p[i], with p a prime of the form 4k+1. Since this ring is commutative with unity and is not a field, it is feasible to study various classes of special elements such as invertibles, zero-divisors, idempotents, and nilpotents. The method for this study consists of developing computer programs in Python, through which the lists of special elements in  Z_p[i] are generated for different values of p. The patterns that characterize these lists are sought, in addition to the cardinality of each of these sets. Subsequently, conjectures of mathematical type are stated for each of these classes of elements, which reflect the observed patterns and properties. Finally, formal mathematical proofs of all the conjectures found are made based on various concepts and results of the theory of numbers, groups, and rings. Thus, we show that Python programming, properly used as part of a method, becomes an important tool to identify patterns, properties, and characteristics of several abstract concepts, typical of algebra.

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Author Biographies

Jesus Antonio Avila Guzman , Universidad del Tolima

J. Avila. He was born in Líbano, Tolima - Colombia, in 1972. He graduated in Mathematics and Physics from the Universidad del Tolima in 1995. He studied a Master's Degree in Mathematics at the Universidad Nacional de Colombia, Bogotá, and graduated in 2002. PhD. in Mathematics from the Universidade Federal do Rio Grande do Sul, Brazil, and graduated in 2008. He currently works as a professor at the Universidad del Tolima. His areas of interest are algebra and topology

Oscar Eduardo Trujillo-niño , University of Puerto Rico-Mayaguez

O. E. Trujillo-Niño. He was born in Ibagué, Tolima-Colombia, in 1996. He graduated in Mathematics with Emphasis in Statistics from the Universidad del Tolima in 2022. He is currently studying for a master's degree in mathematicsstatistics at the University of Puerto Rico, Mayagüez campus. He is also working as a data scientist in the company Lumni Colombia S.A.

José David Liévano-González , University of Tolima

J. D. Liévano-González. He was born in Ibagué, Tolima - Colombia, in 1993. He completed undergraduate studies at the Universidad del Tolima, obtaining a degree of Professional in Mathematics with Emphasis in Statistics in 2021. Subsequently, he took certified courses in software development and data science. Currently, he works as a data scientist in the collection management industry.

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Published

2025-03-31

How to Cite

Avila Guzman , J. A., Trujillo-niño , O. E., & Liévano-González , J. D. (2025). Python programming and algebra: Some special elements in Gaussian integers modulo a prime . Scientia Et Technica, 30(01), 48–57. https://doi.org/10.22517/23447214.25748

Issue

Vol. 30 No. 01 (2025): Enero-Marzo

Section

Ciencias Básicas

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