Existence results for primitive elements in cubic and quartic extensions of a finite field
Geoff B Bailey, Stephen D. Cohen, Nicole Sutherland, Tim Trudgian · Mathematics of Computation · 2018
With F q \mathbb {F}_q the finite field of q q elements, we investigate the following question. If γ \gamma generates F q n \mathbb {F}_{q^n} over F q \mathbb {F}_q and if β \beta is a nonzero element of F q n \mathbb {F}_{q^n} , is there always an a ∈ F q a \in \mathbb {F}_q such that β ( γ + a ) \beta (\gamma + a) is a primitive element? We resolve this case when n = 3 n=3 , thereby proving a conjecture by Cohen. We also substantially improve on what is known when n = 4 n=4 .