Irreducible Polynomials in $\mathbb F_{11}$

Question

I'm searching all irreducible polynomials in $\mathbb{F}_{11} h(x) = x^3 + \ldots$

What is the fastest way to get it? I tried to factorize $x^{11\times3}-x$ but without success. Any advice?

Answer 1

There $440$ (only the monic) of such polynomials. I don't think you want to create a list of them. A way to describe them would be the following:

Describe the reducible polynomials. They are of the form $(x-a)(x-b)(x-c)$ with $a,b,c \in \mathbb F_{11}$ or of the form $(x-a)((x-b)^2-c)$ with $a,b \in \mathbb F_{11}, c \in \{2,6,7,8,10\}$ (The non-squares).