Abstract

Let $F={\mathbb{F}}_{q^m},\hspace{0.5em}m\ge 7,$ n a positive integer, and $f=p_1/p_2$ with p ₁, p ₂ co-prime irreducible polynomials in $F[x]$ and $\mathrm{deg}(p_1)+\mathrm{deg}(p_2)=n.$ We obtain a sufficient condition on (q, m), which guarantees, for any prescribed a, b in $E={\mathbb{F}}_q,$ the existence of primitive pair $(\alpha ,f(\alpha ))$ in F such that ${\mathrm{}\text{Tr}}_{F/E}(\alpha )=a$ and ${\mathrm{}\text{Tr}}_{F/E}({\alpha }^{-1})=b.$ Further, for every positive integer n, such a pair definitely exists for large enough m. The case n = 2 is dealt separately and proved that such a pair exists for all (q, m) apart from at most 64 choices.

摘要

让 $F={\mathbb{F}}_{q^{米}}，\hspace{0.5em}米\ge 7，$ n为正整数，并且$F=p_{\mathrm{1个}}/p_2$与p ₁，p ₂的素数不可约多项式$F[X]$ 和 $度（p_{\mathrm{1个}}）+度（p_2）=ñ。$我们获得（一个充分条件q，米），它的保证，对于任何规定的一个，b中$Ë={\mathbb{F}}_q，$ 原始对的存在 $（\alpha ，F（\alpha ））$在F中${\mathrm{}\text{Tr}}_{F/Ë}（\alpha ）=\mathrm{一种}$ 和 ${\mathrm{}\text{Tr}}_{F/Ë}（{\alpha }^{-\mathrm{1个}}）=b。$此外，对于每个正整数n，对于足够大的m，肯定存在这样的一对。 单独处理n = 2的情况，证明除了最多64个选择之外，所有（q，m）都存在这样的一对。