I am currently assisting a course for future teachers at university level for a joint education in maths and physics in Germany and I have a question regarding the use and possible application of the following result.
We defined the annihilator for an element A of a K-algebra $\mathcal A$ as the set of polynomials over the field K, that return zero, i.e.
$$Ann(A):=\{f\in K[x]: \tilde f(A)=0\}$$
The tilda indicates the evaluation-homomorphism. We proved that any such f can be decomposed into a unique normed polynomial h and some rest, such that every $f = g\cdot h$. This can be used to show that a nilpotent matrix of order k, has $M_A(x)=x^k$ as minimal polynomial.
My problem: it is an interesting tool to find the minimal polynomial of say a real valued matrix. But I do not see the relevance for future teachers as this seems to be "just" a technical result without any applications that one could explain to someone without extensive mathematical education. I do know, that the subspace of linear functions with the same property is a subspace of the dual space.
So, are there any nice applications for this method in the realm of euclidean geometry? or physics? or maths?
If you feel this is not the right place to ask such an involved question, feel free to suggest an alternative. I have posted this very question to math.stackexchange, too.