In this article, we establish a sufficient condition for the existence of aprimitive element $\alpha \in {\mathbb{F}_{q^n}}$ such that the element$\alpha+\alpha^{-1}$ is also a primitive element of ${\mathbb{F}_{q^n}},$ and$Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in\mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. Weprove that every finite field $\mathbb{F}_{q^n}~ (n \geq5),$ contains suchprimitive elements except for finitely many values of $q$ and $n$. Indeed, bycomputation, we conclude that there are no actual exceptional pairs $(q,n)$ for$n\geq5.$
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机译：在本文中，我们为{\ mathbb {F} _ {q ^ n}} $中原始元素$ \ alpha \的存在建立了充分条件，使得元素$ \ alpha + \ alpha ^ {-1} $为也是$ {\ mathbb {F} _ {q ^ n}}，$和$ Tr _ {\ mathbb {F} _ {q ^ n} | \ mathbb {F} _ {q}}（\ alpha ）= a $对于任何规定的$ a \ in \ mathbb {F} _q $，其中$ q = p ^ k $对于一些质数$ p $和正整数$ k $。我们证明每个有限域$ \ mathbb {F} _ {q ^ n}〜（n \ geq5），$都包含这样的本原元素，除了$ q $和$ n $的数量有限。实际上，通过计算，我们得出结论，对于$ n \ geq5。$，没有实际的例外对$（q，n）$。
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