Generating the Group of Nonzero Elements of a Quadratic Extension of $\mathbb{F}_{p}$

TL;DR

This paper characterizes generators of the multiplicative group of a quadratic extension of a finite field, providing necessary and sufficient conditions involving the norm map and its kernel.

Contribution

It introduces precise criteria for identifying generators of the group in quadratic extensions of finite fields, expanding understanding of their structure.

Findings

Generators are characterized by norm and kernel conditions.

Provides a method to determine non-generators of the kernel.

Establishes necessary and sufficient conditions for generator identification.

Abstract

It is well known that if $\mathbb{F}$ is a finite field then $\mathbb{F^{*}}$, the set of non zero elements of $\mathbb{F}$, is a cyclic group. In this paper we will assume $\mathbb{F}=\mathbb{F}_{p}$ (the finite field with p elements, p a prime) and $\mathbb{\mathbb{F}}_{p^{2}}$ is a quadratic extension of $\mathbb{F}_{p}$. In this case, the groups $\mathbb{F}_{p}^{*}$ and $\mathbb{F}_{p^{2}}^{*}$ have orders $p-1$ and $p^{2}-1$ respectively. We will provide necessary and sufficient conditions for an element $u\in\mathbb{F}_{p^{2}}^{*}$ to be a generator. Specifically, we will prove $u$ is a generator of $\mathbb{F}_{p^{2}}^{*}$ if and only if $N(u)$ generates $\mathbb{F}_{p}^{*}$ and $\frac{u^{2}}{N(u)}$ generates Ker$\,N$, where $N:\mathbb{F}_{p^{2}}^{*}\rightarrow\mathbb{F}_{p}^{*}$ denotes the norm map. We will also provide a method for determining if $u$ is not a generator of Ker$\,N$.