Say $T:\ V\to V$ is a linear operator And $V=Z_1\oplus Z_2\oplus Z_3 \oplus \cdots \oplus Z_S$ is a $T$-cyclic decomposition of $V$ where $Z_i=Z(T,v_i)$. Now, we assume that $p^e(X)\mid M_T(X)$ where $p(X)$ is an irreducible polynomial of degree $d$.
I'm trying to proof that $p(T)^e(V)/p(T)^{e+1}(T)(V)$ is isomorphic to the direct sum $\bigoplus^s_{i=1}[p(T)^e(V)\cap Z_i]/[p(T)^{e+1}(T)(V)\cap Z_i]]$.
I have proved that $p(T)^e(V)\cap Z_i=Z(T,p(TP^e(v_i))=p(T)^e(Z(T,v_i)).$ And then the direct sum part is $d$-dimensional vector space iff $p(X)^{e+1}\mid M_{t,V_i}(X).$ What should I do next?