On the Primitivity of Trinomials over Small Finite Fields.

In this paper, we explore the primitivity of trinomials over small finite fields. We extend the results of the primitivity of trinomials x + ax + b over F4 [1] to the general form x + ax + b. We prove that for given n and k, one of all the trinomials x + ax + b with b being the primitive element of F4 and a + b 6= 1 is primitive over F4 if and only if all the others are primitive over F4. And we can deduce that if we find one primitive trinomial over F4, in fact there are at least four primitive trinomials with the same degree. We give the necessary conditions if there exist primitive trinomials over F4. We study the trinomials with degrees n = 4 +1 and n = 21 ·4+29, where m is a positive integer. For these two cases, we prove that the trinomials x+ax+b with degrees n = 4 + 1 and n = 21 · 4 + 29 are always reducible if m > 1. If some results are obviously true over F3, we also give it.