Rational canonical form with irreducible minimal polynomial ($m_A=f\text{ irreducible}⟹A\sim C_f^{\oplus n/\mathrm{deg}f}$)

Let $A$ act on a nonzero finite-dimensional vector space over a field $F$, with minimal polynomial an irreducible polynomial $f$ of polynomial degree $d$. The nonconstant invariant factors of a linear operator all divide $f$, so each equals $f$. Their degrees sum to the dimension of a vector space, giving $d\mid n$ and rational canonical form consisting of $n/d$ copies of the companion matrix $C_f$. Equivalently, make the vector space a module over the field $F[X]/(f)$ by letting $X$ act as $A$; a basis over this field gives the same block decomposition.