Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 2 / b / i / Solution

The primary decomposition theorem for linear maps states that ifwith distinct monic irreducible polynomials $p_j$, thenEach 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.