coprime factorization and CRT

When the minimal polynomial of T factors into pieces that share no common factor, the whole space V breaks apart along those pieces. This is the operator-theoretic face of the Chinese remainder theorem: coprime factors of m(x) split V into a direct sum of invariant subspaces, one per factor.

Concretely, suppose m(x) = f(x) g(x) with f and g coprime (gcd(f, g) = 1). Then V = ker f(T) (+) ker g(T) as a direct sum of T-invariant subspaces. The mechanism is the Bezout identity: coprimeness gives polynomials a, b with a f + b g = 1, so for any v you get v = a(T) f(T) v + b(T) g(T) v, and the two pieces land in ker g(T) and ker f(T) respectively — an explicit splitting built from polynomials in T.

Iterating this on the full factorization m(x) = p_1(x)^e_1 ... p_k(x)^e_k into coprime prime powers gives the primary decomposition: V = (+) ker p_i(T)^e_i, where each summand is the generalized eigenspace when p_i is linear. The minimal polynomial of T restricted to the i-th summand is exactly p_i^e_i, so the operator decouples into independent, simpler pieces you can analyze one at a time.

This is the structural workhorse behind everything downstream. Diagonalizability (distinct linear factors -> each piece is a scalar block), Jordan form (linear factors over C, each piece a nilpotent-plus-scalar), and the nilpotent-plus-semisimple decomposition all flow from splitting V by the coprime factors of its minimal polynomial. Algebra (factoring a polynomial) becomes geometry (splitting a space).