It is well known that if F is a finite field then
Fβ, the set of non zero elements of F, is a cyclic
group. In this paper we will assume F=Fpβ (the finite
field with p elements, p a prime) and Fp2β is a
quadratic extension of Fpβ. In this case, the groups
Fpββ and Fp2ββ have orders pβ1 and
p2β1 respectively. We will provide necessary and sufficient conditions for
an element uβFp2ββ to be a generator. Specifically, we
will prove u is a generator of Fp2ββ if and only if N(u)
generates Fpββ and N(u)u2β generates KerN,
where N:Fp2βββFpββ denotes the norm
map. We will also provide a method for determining if u is not a generator of
KerN