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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi Jurnal Matematika Integratif
Faldiyan, Muhammad
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Computer Science & IT Control & Systems Engineering Decision Sciences, Operations Research & Management Economics, Econometrics & Finance Electrical & Electronics Engineering Engineering Mathematics Mechanical Engineering Transportation

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Characteristic of Quaternion Algebra Over Fields Faldiyan, Muhammad; Carnia, Ema; Supriatna, Asep K.
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 7, No 4 (2023): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v7i4.17625

Abstract

Quaternion is an extension of the complex number system. Quaternion are discovered by formulating 4 points in 4-dimensional vector space using the cross product between two standard vectors. Quaternion algebra over a field is a 4-dimensional vector space with bases  and the elements of the algebra are members of the field. Each element in quaternion algebra has an inverse, despite the fact that the ring is not commutative. Based on this, the purpose of this study is to obtain the characteristics of split quaternion algebra and determine how it interacts with central simple algebra. The research method used in this paper is literature study on quaternion algebra, field and central simple algebra. The results of this study establish the equivalence of split quaternion algebra as well as the theorem relating central simple algebra and quaternion algebra. The conclusion obtained from this study is that split quaternion algebra has five different characteristics and quaternion algebra is a central simple algebra with dimensions less than equal to four.
Application of Module Structure to Coding Theory: A Systematic Literature Review Faldiyan, Muhammad; Sylviani, Sisilia
Jurnal Matematika Integratif Vol 21, No 1: April 2025
Publisher : Department of Matematics, Universitas Padjadjaran

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24198/jmi.v21.n1.60276.103-112

Abstract

A systematic literature review is a research process that identifies, evaluates, and interprets all relevant study findings connected to specific research questions, topics, or phenomena of interest. In this work, a thorough review of the literature on the issue of the link between module structure and coding theory was done. A literature search yielded 470 articles from the Google Scholar, Dimensions, and Science Direct databases. After further article selection process, 14 articles were chosen to be studied in further depth. The items retrieved were from the previous ten years, from 2012 to 2022. The PRISMA analytical approach and bibliometric analysis were employed in this investigation. A more detailed description of the PRISMA technique and the significance of the bibliometric analysis is provided. The findings of this study are presented in the form of brief summaries of the 14 articles and research recommendations. At the end of the study, recommendations for future development of the code structure utilized in the articles that are further investigated are made
Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields Faldiyan, Muhammad; Carnia, Ema; Supriatna, Asep Kuswandi
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 8, No 2 (2023): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v8i2.22881

Abstract

Quaternions are extensions of complex numbers that are four-dimensional objects. Quaternion consists of one real number and three complex numbers, commonly denoted by the standard vectors  and . Quaternion algebra over the field is an algebra in which the multiplication between standard vectors is non-commutative and the multiplication of standard vector with itself is a member of the field. The field considered in this study is the quadratic field and its extensions are biquadratic and composite. There have been many studies done to show the existence of split properties in quaternion algebras over quadratic fields. The purpose of this research is to prove a theorem about the existence of split properties on three field structures, namely quaternion algebras over quadratic fields, biquadratic fields, and composite of  quadratic fields. We propose two theorems about biquadratic fields and composite of  quadratic fields refer to theorems about the properties of the split on quadratic fields. The result of this research is a theorem proof of three theorems with different field structures that shows the different conditions of the three field structures. The conclusion is that the split property on quaternion algebras over fields exists if certain conditions can be met.
Co-Authors Asep K. Supriatna Asep Kuswandi Supriatna Ema Carnia Sisilia Sylviani