= Rational canonical form with irreducible minimal polynomial
{title2=$m_A=f\text{ irreducible}\Longrightarrow A\sim C_f^{\oplus n/\deg f}$}
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.
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