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.
Linear combinations of primitive elements of a finite field
