= Solution
Make $V$ a $k[x]$-module by $xv=\varphi(v)$. The structure theorem over the Euclidean domain $k[x]$ decomposes its torsion module as
$$
V\cong\bigoplus_\alpha k[x]/(p_\alpha^{e_\alpha}),
$$
where the $p_\alpha$ are monic irreducibles. Each summand is indecomposable: its submodules form a chain, so two nonzero submodules cannot form a direct sum. This is the desired decomposition into invariant indecomposable subspaces.
The <characteristic polynomial> is
$$
\chi_\varphi(x)=\prod_\alpha p_\alpha(x)^{e_\alpha},
$$
whereas the <minimal polynomial> is the least common multiple
$$
m_\varphi(x)=\operatorname{lcm}_\alpha p_\alpha(x)^{e_\alpha}.
$$
Solved by gpt-5.6-sol high.
Back to article page
