I have to find the characteristic polynomial, the minimal polynomial, the invariant factors and the elementary divisors of $M$, where $M=K^3$ is a $K[T]$-module for some field $K$ and $T$ acts as the matrix \begin{equation} \left( \begin{array}{ccc} 0&1&0 \\ 0&0&0\\ 0&0&1\\ \end{array} \right) \end{equation}
In the problem above, I was able to find all the answers, but, being honest, I am not sure about them. The characteristic polynomial of $T$ is $\mathrm{char}_T(x) = \det(xI-T) = x^2(x-1)$. The minimal polynomial must have the same roots, thought, it must be $x(x-1)$ or $x^2(x-1)$, but with minimal degree such that $\min_T(T)=0$. Since $T(T-1) \neq 0$ and $T^2(T-1)=0$, we conclude that $\min_T(x)=x^2(x-1)$. This way, by the Structure Theorem, $$M \simeq K^3[x]/\langle x^2(x-1) \rangle.$$
My bigger problem is finding the invariant factors and the elementary divisors. Could someone give me a hand?
Invariant factors: only one $x^2(x-1)$.
Applying Chinese remainder theorem:
$$ K[x]/\left<x^2(x-1)\right>\cong K[x]/x^2\oplus K[x]/(x-1). $$
Then the elementary divisors are $\{x^2, x-1\}$.