I came across the following problem in my advanced Linear Algebra class, and couldn't do much to solve it. I personally have a bit of a hard time with these structure/decomposition theorems for linear operators, so my ideas quickly become very messy and lead to dead ends...
Let $v \in V$ be a vector with minimal annihilating polynomial $m_{T, v}(x) = p^m(x)$, where $p(x) \in K[x]$ is irreducible. Prove that the only $T$-cyclic subspaces of $Z(v, T)$ are of the form $Z(p^k(T)(v), T)$, where $0 \leq k \leq m$.
If $w \in Z(v, T)$, then $w = f(T)(v)$, for $f(x) \in K[x]$. Thus, by the Fundamental Theorem of Arithmetic for Polynomials, we get that $w = u(T)p^k(T)(v)$, where $(u, p) = 1$ and $k \in \{0, ..., m\}$.
It is therefore clear that $Z(w, T) \subset Z(p^k(T)(v), T)$. But I can't get the other inclusion to work. My main attempt was to write $y \in Z(p^k(T)(v), T)$ as $y = g(T)p^k(T)(v)$ and then divide $g(x)$ by $u(x)$. If the remainder were $0$, we would be done, but I couldn't work out the case where the remainder is some non-zero polynomial. I guess this would use the "co-primality" of $u$ and $p$, but, again, I couldn't really develop it.
Could anyone please give me some hints as per how to proceed?
Thanks in advance!
EDIT: As has been requested, here are some explanations of the notation: $Z(v, T) = \{f(T)(v) \mid f(x) \in K[x]\}$, the set of images of $v$ by polynomials in $T$ and a "$T$-cyclic subspace" is a subspace of the form $Z(w, T)$ for some $w \in V$.