论文标题
原始元素对有限场的立方扩展中有规定的痕迹
Primitive element pairs with a prescribed trace in the cubic extension of a finite field
论文作者
论文摘要
我们证明，对于任何主要功率$ q \ notin \ {3,4,5 \} $，Cutic Extension $ \ Mathbb {f} _ {q^3} $ $ \ textrm {tr} _ {\ mathbb {f} _ {q^3}/\ mathbb {f} _q} _q}（ξ）= a $对于任何处方的$ a \ in \ mathbb {f} _q $。这完成了Gupta，Sharma和Cohen的猜想的证明，这些证明是在任意程度$ n \ ge3 $的延伸期间就类似问题的证明。
We 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 $ξ$ such that $ξ+ξ^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(ξ)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$.
