The unpublished preprint:

D. G. A. Jackson, The irreducibility of a cubic over $\mathbb{F}_q$, Research Report 98-17, Univ. of Sydney (1998)

gives necessary and sufficient conditions (when ${\rm char}({\mathbb F}_q)\neq2,3$) for a cubic polynomial over $\mathbb{F}_q$ to be irreducible.
I have conditions (below) which work for cubics when ${\rm char}(\mathbb{ F}_q)=2,3$, and also quartics. While I guess that such results have been published, I have been unable to find references. I would be most grateful if someone knows precise reference (if they exist).

**Theorem A.** Suppose $q=2^e$ and $c(x)=x^3+c_2x+c_3$ where $c_2c_3\neq0$. Then $c(x)$ is irreducible over $\mathbb{F}_q$ if and only if ${\rm Tr}_{\mathbb{F}_q/\mathbb{F}_2}(c_2^3/c_3^2)\equiv e\pmod2$, and a root $\beta$
of $b(x)=x^2+c_3x+c_2^3$ is a noncube.

**Theorem B.** Suppose $q=3^e$ and $c(x)=x^3+c_2x+c_3$. Then $c(x)$ is irreducible over $\mathbb{F}_q$ if and only if $-c_2$ is a square in $\mathbb{F}_q^\times,$ and ${\rm Tr}_{\mathbb{F}_q/\mathbb{F}_3}(c_3/\eta^3)\neq0$ where $\eta^2=-c_2$.

The factorization of a quartic should depend on two square roots and one cube root, at least when ${\rm char}(\mathbb{F}_q)\neq2,3$. Given that 1/4 of quartic polynomials are irreducible (in the limit as $q\to\infty$) one may guess that there is an irreducibility test of the form: $x^4+ax^3+bx^2+cx+d$ is irreducible over $\mathbb{F}_q$ if and only if $f$ or $g$ is a square, and $h$ is a cube; where $f,g,h$ are given rational functions of $a,b,c,d$. [Thus irreducibility should occur with limiting probability $(1-(1/2)^2)\times 1/3=1/4$, as $q\to\infty$; agreeing with the correct limiting probability.]

**Question.** Are expressions for $f,g,h$ known? What if ${\rm char}(\mathbb{F}_q)=2,3$? [I should say: "f is a square or g is a nonsquare". The probabilistic argument still works.]

Stephen Glasby