= Solution
The annihilating polynomial excludes any free $\mathbb C[X]$-summand. In a cyclic <primary decomposition> summand $\mathbb C[X]/(P^a)$, the same condition requires $P^a$ to divide $(X-2)^4$. Thus $P=X-2$, up to a scalar unit, and $1\leq a\leq4$. The complex <dimension> of this summand is $a$.
Write $A_j=\mathbb C[X]/((X-2)^j)$. The exponents must form a <partition of an integer> $4$, giving exactly
$$
\boxed{A_4,\quad A_3\oplus A_1,\quad A_2\oplus A_2,\quad A_2\oplus A_1\oplus A_1,\quad A_1^{\oplus4}.}
$$
They are distinct <module isomorphism> types by uniqueness in the <structure theorem for finitely generated modules over a principal ideal domain>; equivalently they are the five possible <Jordan block> patterns for eigenvalue $2$ in dimension four.
Back to article page
