FINITE FIELD EXTENSIONS WITH THE LINE OR TRANSLATE PROPERTY FOR -PRIMITIVE ELEMENTS
2020 ◽
pp. 1-7
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Keyword(s):
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.
Keyword(s):
2020 ◽
Vol 16
(09)
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pp. 2027-2040
Keyword(s):
2018 ◽
Vol 51
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pp. 388-406
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1980 ◽
Vol 32
(6)
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pp. 1299-1305
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1999 ◽
Vol 45
(7)
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pp. 2601-2605
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