 Nullity of a cyclic linear operator

For a nonzero finite-dimensional cyclic linear operator, identify its module with $K[X]/(d)$, where $d$ is the monic minimal polynomial. Its kernel consists of classes of $h$ with $d\mid Xh$. If $d(0)\ne 0$, coprimality gives $d\mid h$, so the kernel is zero. If $d=Xe$, it consists precisely of the multiples of $e$ modulo $Xe$, a one-dimensional space. Thus the multiplicity of zero as a root of $d$ does not increase the nullity. The same companion block reads off invertibility from the constant term and gives the characteristic and minimal polynomials directly, while explicitly listing eigenvalues may require factoring $d$.