= Solution
The <rational canonical form> of $T$ is a block diagonal matrix
$$
C(f_1)\oplus\cdots\oplus C(f_r),\qquad
f_1\mid f_2\mid\cdots\mid f_r,
$$
similar to a matrix of $T$, where the $f_j\in F[t]$ are monic nonconstant polynomials and the sum of their degrees is $\dim V$. They are the <invariant factors of a linear operator>. The block $C(f_j)$ is the <companion matrix> for $f_j$, representing multiplication by $t$ on $F[t]/(f_j)$. The invariant factors are uniquely determined by $T$.
Every companion block has a <cyclic vector> and therefore has both <minimal polynomial> and <characteristic polynomial> equal to $f_j$, by part (a). The <determinant of a block diagonal matrix> is the product of its block determinants, giving
$$
\boxed{c_T(t)=\prod_{j=1}^r f_j(t).}
$$
A polynomial $h$ annihilates the whole matrix exactly when it annihilates every block. On block $j$, dividing $h$ by $f_j$ leaves a remainder of smaller degree; if that remainder annihilated the block without being zero, it would contradict minimality of $f_j$. Thus the annihilation condition is exactly $f_j\mid h$ for every $j$. The monic polynomial of least degree with this property is their <least common multiple>. The divisibility chain makes this $f_r$, so
$$
\boxed{m_T(t)=f_r(t).}
$$
These are the <minimal and characteristic polynomials from invariant factors> formulae. On the zero vector space the product is empty and both polynomials are defined to be one.
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