Paper 2014/659

On the Primitivity of Trinomials over Small Finite Fields

YUjuan Li, 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.