= Solution
The <primary decomposition theorem for linear maps> states that if
$$
m_T(x)=\prod_{j=1}^r p_j(x)^{e_j}
$$
with distinct monic <irreducible polynomials> $p_j$, then
$$
\boxed{V=\bigoplus_{j=1}^r\ker\bigl(p_j(T)^{e_j}\bigr).}
$$
Each summand is a $T$-<invariant subspace>. More generally the same <direct sum> conclusion holds for any factorization of an annihilating <polynomial> into pairwise <coprime polynomials>, using the corresponding kernels. The underlying mechanism is the <Bézout's identity for polynomials>, which separates the mutually coprime factors.
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