polynomial evaluation at an operator

Fix an operator T and consider the rule that takes a polynomial p(x) and produces the operator p(T): replace every x by T, every constant c by cI, and read the result as a single operator. For example x^2 + 3x - 5 becomes T^2 + 3T - 5I. This rule is the evaluation map ev_T: F[x] -> End(V).

The magic is that ev_T respects all the arithmetic. It sends sums to sums, scalar multiples to scalar multiples, and — the key fact — products to products: (pq)(T) = p(T) q(T). It also sends the constant polynomial 1 to the identity I. A map between rings that preserves addition, multiplication, and the unit is a ring homomorphism, so ev_T is a homomorphism of algebras from F[x] into End(V).

Its image is F[T], the subalgebra of all polynomials in T, which is automatically COMMUTATIVE — even though End(V) at large is not — because polynomials in the single element T commute with one another. Its kernel is the set of polynomials sent to the zero operator, i.e. exactly the annihilator ideal I(T) from the previous terms.

Now the first isomorphism theorem for rings closes the loop: F[x] / ker(ev_T) is isomorphic to the image. Since the kernel is generated by the minimal polynomial m, this reads F[x]/(m) is isomorphic to F[T]. Every structural fact about the algebra of T is thus a fact about the quotient ring F[x]/(m) — a beautifully clean transfer from operators to polynomials.