**On the Primitivity of Trinomials over Small Finite Fields**

*YUjuan Li and Jinhua Zhao and Huaifu Wang*

**Abstract: **In this paper, we
explore the primitivity of trinomials over small finite fields. We
extend the results of the primitivity of trinomials $x^{n}+ax+b$
over ${\mathbb{F}}_{4}$ \cite{Li} to the general form
$x^{n}+ax^{k}+b$. We prove that for given $n$ and $k$, one of all the trinomials
$x^{n}+ax^{k}+b$ with $b$ being the primitive element of
${\mathbb{F}}_{4}$ and $a+b\neq1$ is primitive over
${\mathbb{F}}_{4}$ if and only if all the others are primitive over
${\mathbb{F}}_{4}$. And we can deduce that if we find one primitive
trinomial over ${\mathbb{F}}_{4}$, in fact there are at least four primitive
trinomials with the same degree. We give the necessary conditions if
there exist primitive trinomials over ${\mathbb{F}}_{4}$. We study
the trinomials with degrees $n=4^{m}+1$ and $n=21\cdot4^{m}+29$,
where $m$ is a positive integer. For these two cases, we prove that
the trinomials $x^{n}+ax+b$ with degrees $n=4^{m}+1$ and
$n=21\cdot4^{m}+29$ are always reducible if $m>1$. If some results
are obviously true over ${\mathbb{F}}_{3}$, we also give it.

**Category / Keywords: **

**Date: **received 24 Aug 2014

**Contact author: **liyj at amss ac cn

**Available format(s): **PDF | BibTeX Citation

**Version: **20140827:074834 (All versions of this report)

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