$r$ -本原元具有行或平移性质的有限域扩张
Finite field extensions with the line or translate property for $r$-primitive elements
摘要：
令$r,n>1$为整数，$q$为任意素数幂$q$，使得$r\mid q^n-1$。我们说扩张$\mathbb{F}_{q^n}/\mathbb{F}_q$具有$r$的线性质--本原元性质，如果对每个$α,θ\in\mathbb{F}_{q^n}^*$，使得$\mathbb{F}_{q^n}=\mathbb{F}_q(θ)$，存在一些$x\in\mathbb{F}_q$，使得$α(θ+x)$具有乘法阶$(q^n-1)/r$ .我们证明了，对于充分大的素数幂$q$，$\mathbb{F}_{q^n}/\mathbb{F}_q$具有$r$ -本原元的直线性质.我们还讨论了（较弱的）平移性质的扩展。
英文摘要：
Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^n-1$. We say that the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the line property for $r$-primitive elements property if, for every $α,θ\in\mathbb{F}_{q^n}^*$, such that $\mathbb{F}_{q^n}=\mathbb{F}_q(θ)$, there exists some $x\in\mathbb{F}_q$, such that $α(θ+x)$ has multiplicative order $(q^n-1)/r$. We prove that, for sufficiently large prime powers $q$, $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the line property for $r$-primitive elements. We also discuss the (weaker) translate property for extensions.
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