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Journals / International Electronic Journal of Geometry / 2026 / Volume: 19 Issue: 1

The Number of $k$-potent Elements in the Quaternion Algebra $\mathbb{H}_{\mathbb{Z}_{p}}$

Cristina Flaut(Ovidius University)0000-0003-2714-0583
Andreea Baias(Ovidius University of Constanta, Romania)0009-0004-8162-6212
Pages
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Publication Date
2026-04-22
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

For this purpose, we will use computer as a tool to check and understand the behavior of these elements in all cases that will be studied.

📌 3 Key Takeaways

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Objective & Hypothesis:In this paper we count the number of $k$ -potent elements over $\mathbb{H}_{\mathbb{Z}_{p}}$ ,where $\mathbb{H}_{\mathbb{Z}_{p}}$ is the quaternion algebra over $\mathbb{Z}_{p}$ , and we present a descriptive formula for the general case.
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Key Findings & Data:For $k\in \{3,4,5\}$ , we give an explicit formula for these values.
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Conclusion & Impact:For this purpose, we will use computer as a tool to check and understand the behavior of these elements in all cases that will be studied.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Quaternions · $k$-potent elements over $\mathbb{H}_{\mathbb{Z}_{p}}$ · quaternion algebra over $\mathbb{Z}_{p}$

Abstract

In this paper we count the number of $k$ -potent elements over $\mathbb{H}_{\mathbb{Z}_{p}}$ ,where $\mathbb{H}_{\mathbb{Z}_{p}}$ is the quaternion algebra over $\mathbb{Z}_{p}$ , and we present a descriptive formula for the general case. For $k\in \{3,4,5\}$ , we give an explicit formula for these values. Moreover, as an application of these results, we count the number of solutions of the equation $x^{k}=1$ over $\mathbb{H}_{ \mathbb{Z}_{p}}$. For this purpose, we will use computer as a tool to check and understand the behavior of these elements in all cases that will be studied.

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Cite This Article

Flaut, C., Baias, A. (2026). The Number of $k$-potent Elements in the Quaternion Algebra $\mathbb{H}_{\mathbb{Z}_{p}}$. International Electronic Journal of Geometry, Volume: 19 Issue: 1. https://doi.org/10.36890/iejg.1759957

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