PRIMITIVE ELEMENT PAIRS WITH A PRESCRIBED TRACE IN THE CUBIC EXTENSION OF A FINITE FIELD
Andrew R. Booker, Stephen D. Cohen, Nicol Leong, Tim Trudgian · Bulletin of the Australian Mathematical Society · 2022
Abstract We prove that for any prime power $q otin \{3,4,5\}$ , the cubic extension $\mathbb {F}_{q^{3}}$ of the finite field $\mathbb {F}_{q}$ contains a primitive element $\xi $ such that $\xi +\xi ^{-1}$ is also primitive, and $\operatorname {\mathrm {Tr}}_{\mathbb {F}_{q^{3}}/\mathbb {F}_{q}}(\xi )=a$ for any prescribed $a\in \mathbb {F}_{q}$ . This completes the proof of a conjecture of Guptaet al.[‘Primitive element pairs with one prescribed trace over a finite field’,Finite Fields Appl.54(2018), 1–14] concerning the analogous problem over an extension of arbitrary degree $n\ge 3$ .