= Solution
The \b[<minimal polynomial>] $m_T$ is the monic <polynomial> of least <polynomial degree> such that $m_T(T)=0$. Here a <polynomial> $p(x)=\sum_jc_jx^j$ is evaluated at the <linear operator> by $p(T)=\sum_jc_jT^j$, with the constant term acting as $c_0I$.
An annihilating <polynomial> exists because $I,T,T^2,\ldots$ lie in the finite-dimensional <vector space> $\operatorname{End}_F(V)$ and are therefore eventually <linearly dependent>. Uniqueness follows by subtracting two monic annihilators of the same least degree: their difference has smaller degree and still annihilates $T$.
Equivalently,
$$
\boxed{m_T\mid p\quad\Longleftrightarrow\quad p(T)=0.}
$$
For the forward implication evaluate the factorization at $T$. For the reverse implication divide $p$ by $m_T$ using <polynomial division>; the remainder annihilates $T$, so minimality makes it zero. This also identifies the annihilating polynomials as the <principal ideal> generated by $m_T$.
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