This is a question related to the proof of the rational canonical form (Frobenius canonical form).
Suppose $T$ is an operator on $V$ and $V=\ker(p^d(T))$ where $p$ is an irreducible polynomial. Then I want to show that there exists $w_1,\cdots,w_n$ such that $$ w_1,Tw_1,\cdots,T^{e_1}w_{1};\cdots;w_n,Tw_n,\cdots,T^{e_n}w_{n} $$ is a basis of $V$ and $\sigma d-1=e_1\geq\cdots\geq e_n$, where $\sigma=\deg p$. Then from this I could derive the rational canonical form.
My approach to this lemma is by considering the nilpotent operator $N=p(T)$. Then there exists a cyclic decomposition of $V$ into $N-$cyclic subspaces $$ N^{m_1}v_1,\cdots,Nv_1,v_1;\cdots;N^{m_n}v_n,\cdots,Nv_n,v_n. $$ Assume $d-1=m_1\geq\cdots\geq m_n\geq0$ and let $l_d,\cdots,l_1$ be nonnegative integers $$ m_1=\cdots=m_{l_d}=d-1,\cdots,m_{l_{d}+\cdots+l_2+1}=\cdots=m_{l_{d}+\cdots+l_{1}}=0. $$ In other words, we group the $N-$cyclic subspaces by their dimensions. Then I wish to show that
CLAIM There exists $u_{i1},\cdots,u_{iL_i}\in\operatorname{span}(v_{l_d+\cdots+l_{i+1}+1},\cdots,v_{l_d+\cdots+l_{i}})$ such that $$ V_i=Z(v_{l_d+\cdots+l_{i+1}+1};N)\oplus\cdots\oplus Z(v_{l_d+\cdots+l_{i}};N)=Z(u_{i1};T)\oplus\cdots\oplus Z(u_{iL_i};T) $$ for each $i=d-1,\cdots,1$.
Here, we use $Z(w;L):=\operatorname{span}(w,Lw,\cdots)$ to denote the corresponding cyclic subspaces, and it could be shown that $Z(u_{ik};T)=\operatorname{span}(u_{ik},\cdots,T^{\sigma i-1}u_{ik})$.
I wonder how my claim could be proved (using induction where the base is $i=d$), or is this conjecture not true? If the conjecture is not true, then is there a way for us to prove the proposition by considering the nilpotent operator $N=p(T)$?
The proof is much easier if you use module theory.
Here is a document that teaches you the basics of module theory and then tells you how to arrive at the rational canonical form.
http://buzzard.ups.edu/courses/2014spring/420projects/math420-UPS-spring-2014-toomey-rational-canonical-form.pdf