I've got a problem with understanding the following citation from XOR-counts and lightweight multiplication with fixed elements in binary finite fields on page 10 in the beginning.
Recall that a matrix $M \in GL(n,\mathbb{F}_2)$ is similar to its (unique) rational canonical form. If $M$ has an irreducible minimal polynomial $m$ with $deg(m) = k$ then there exists a $d ≥ 1$ so that $kd = n$ and the rational canonical form is $\bigoplus ^d _{i=1} C_m$ where $C_m$ is a companion matrix for the minimal polynomial and $\bigoplus$ denotes block diagonal matrix.
But that would mean that the degree of the minimal polynomial would be a divisor of $n$. But that's not always the case. I found a counterexample on Wikipedia in Frobenius normal form where $ n=8$ and $deg(m) = 6$, so no such $k \in \mathbb{N}$ exists.
Does that mean that the property mentioned in the article is caused by the fact that we consider a regular matrix or a fact that we consider a matrix over $\mathbb{F}_2$? Thank you.