November 2023 Generating the Group of Nonzero Elements Of a Quadratic Extension Of Fp
Jerry D Rosen, Daniel Sarian, Susan Elizabeth Slome
Missouri J. Math. Sci. 35(2): 248-253 (November 2023). DOI: 10.35834/2023/3502248

Abstract

It is well known that if F is a finite field then F∗, the set of nonzero elements of F, is a cyclic group. In this paper we will assume F = Fp (the finite field with p elements, p a prime) and Fp2 is a quadratic extension of Fp. In this case, the groups F∗p and F∗ p2 have orders p −1 and p2 −1 respectively. We will provide necessary and sufficient conditions for an element u ∈ F∗ p2 to be a generator. Specifically, we will prove u is a generator of F∗ p2 if and only if N(u) generates F∗p and u2 N(u) generates Ker N, where N : F∗ p2 →F∗p denotes the norm map

Citation

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Jerry D Rosen. Daniel Sarian. Susan Elizabeth Slome. "Generating the Group of Nonzero Elements Of a Quadratic Extension Of Fp." Missouri J. Math. Sci. 35 (2) 248 - 253, November 2023. https://doi.org/10.35834/2023/3502248

Information

Published: November 2023
First available in Project Euclid: 28 November 2023

Digital Object Identifier: 10.35834/2023/3502248

Subjects:
Primary: 11A07
Secondary: 12F99

Keywords: finite fields , norm , primitive roots , quadratic residues

Rights: Copyright © 2023 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.35 • No. 2 • Nov 2023
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