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FINITE FIELD EXTENSIONS WITH THE LINE OR TRANSLATE PROPERTY FOR
$r$-PRIMITIVE ELEMENTS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 111 / Issue 3 / December 2021
- Published online by Cambridge University Press:
- 02 March 2020, pp. 313-319
- Print publication:
- December 2021
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- Article
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- You have access
- Open access
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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 
$\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}\in \mathbb{F}_{q^{n}}^{\ast }$ such that 
$\mathbb{F}_{q^{n}}=\mathbb{F}_{q}(\unicode[STIX]{x1D703})$, there exists some 
$x\in \mathbb{F}_{q}$ such that 
$\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}+x)$ 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.
