Monic polynomial with coefficients in $\mathbb F_2$ whose companion matrix is invertible and has largest possible multiplicative order

Question

Let $a_0,...,a_{n-1}\in \mathbb F_2$ be such that the companion matrix of the monic polynomial $a_0+a_1X+\cdots+a_{n-1}X^{n-1}+X^n\in \mathbb F_2[X]$ is invertible and has order $2^n-1$ in $\mathrm{GL}_n(\mathbb F_2)$.

Then is it true that $a_0+a_1X+\cdots+a_{n-1}X^{n-1}+X^n$ is irreducible in $\mathbb F_2[X]$?

Answer 1

$\newcommand{\F}{\mathbb{F}}$Let $f$ be your polynomial, and $C$ its companion matrix.

$f$ is the minimal polynomial of $C$ over $\F_{2}$. Thus the quotient ring $\F_{2}[x] / (f)$, which is a ring of order $2^{n}$, is isomorphic to $\F_{2}[C]$.

You know that $\F_{2}[C]$ contains the multiplicative group of order $2^{n} - 1$ of the powers of $C$. Hence $\F_{2}[C]$ is a field, and so is $\F_{2}[x] / (f)$, and thus $f$ is irreducible in $\F_{2}[x]$.