PRIMITIVE ELEMENT PAIRS WITH A PRESCRIBED TRACE IN THE CUBIC EXTENSION OF A FINITE FIELD

Abstract

AbstractWe prove that for any prime power$q\notin \{3,4,5\}$, 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$.