annihilating polynomial

An annihilating polynomial for an operator T is any polynomial p(x) that kills T when you substitute it in: p(T) = 0, the zero operator. The characteristic polynomial is one (by Cayley-Hamilton), the minimal polynomial is another, and there are infinitely many more — for instance, any multiple of the minimal polynomial.

Collect ALL of them into a set. This set is not random: it is closed under addition (if p(T)=0 and q(T)=0 then (p+q)(T)=0) and, crucially, closed under multiplication by ANY polynomial (if p(T)=0 then (q p)(T) = q(T) p(T) = 0 for every q). A subset of polynomials with those two closure properties is called an ideal. So the annihilating polynomials form the annihilator ideal of T.

Because the polynomial ring F[x] over a field is a principal ideal domain, every ideal is generated by a single element. The monic generator of the annihilator ideal is precisely the minimal polynomial: p annihilates T if and only if the minimal polynomial divides p. That single sentence packages everything — being an annihilator just means being a multiple of m.

This reframing pays off conceptually. Instead of asking 'is p an annihilator?' you ask 'does m divide p?', turning an operator question into a clean divisibility question in F[x]. It is the doorway to viewing V as a module over F[x] and to the whole canonical-form machinery.