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

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.

Keywords

• Quaternions
• $k$-potent elements over $\mathbb{H}_{\mathbb{Z}_{p}}$
• quaternion algebra over $\mathbb{Z}_{p}$

Supporting Institution

Ovidius University of Constanta, Romania

Project Number

no project

Thanks

Authors thank referees for their valuable suggestions who helped us to improve this paper.

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