Past exam of the algebra course of the University of Oxford / 2011 / 3 / a

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.

Apply the decomposition to the module on which $X$ acts as $\alpha$. On $\mathbb{C}[X]/((X-\lambda {)}^m)$ put $t=X-\lambda$ and use the ordered basisMultiplication by $X=\lambda +t$ has diagonal entries $\lambda$ and superdiagonal entries $1$, so its matrix is the Jordan block $J_m(\lambda )$. Concatenating the bases of the direct sum gives a basis of $V$ in whichThis proves existence of Jordan normal form; uniqueness of the primary cyclic decomposition also proves uniqueness of its blocks up to order.