Generating the Group of Nonzero Elements Of a Quadratic Extension Of Fp

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