 Solution

Every finite-dimensional $\mathbb{C}[X]$-module is a finitely generated torsion module over the principal ideal domain $\mathbb{C}[X]$. The structure theorem for finitely generated modules over a principal ideal domain giveswhere the $p_j$ are uniquely determined monic nonconstant polynomials. Equivalently, factoring over $\mathbb{C}$ and using the Chinese remainder theorem givesThere are finitely many summands, and the multisets of positive integers $m_{\lambda j}$ are uniquely determined. Each displayed primary cyclic summand is an indecomposable module.