Given \(m, n, q\in \mathbb {N}\) such that q is a prime power and \(m\ge 3\), \(a\in \mathbb {F}_q\), we establish a sufficient condition for the existence of a primitive pair \((\alpha , f(\alpha ))\) in \(\mathbb {F}_{q^m}\) such that \(\alpha \) is normal over \(\mathbb {F}_q\) and \(\text {Tr}_{\mathbb {F}_{q^m}/\mathbb {F}_q}(\alpha ^{-1})=a\), where \(f(x)\in \mathbb {F}_{q^m}(x)\) is a rational function of degree sum n. Further, when \(n=2\) and \(q=5^k\) for some \(k\in \mathbb {N}\), such a pair definitely exists for all (q, m) apart from at most 20 choices.

给定\(m, n, q\in \mathbb {N}\)使得q是素数幂和\(m\ge 3\) , \(a\in \mathbb {F}_q\)，我们建立在\(\mathbb {F}_{q^m}\) 中存在原始对\((\alpha , f(\alpha ))\)的充分条件使得\(\alpha \)是正常的在\(\mathbb {F}_q\)和\(\text {Tr}_{\mathbb {F}_{q^m}/\mathbb {F}_q}(\alpha ^{-1})= a\)，其中\(f(x)\in \mathbb {F}_{q^m}(x)\)是度和n的有理函数。此外，当\(n=2\)和\(q=5^k\)对于某些\(k\in \mathbb {N}\)，除了最多 20 个选择之外，对于所有 ( q ,  m )肯定存在这样的一对。