Some characterizations about T-indecomposable vector spaces

Question

Recently I have found this statement and I would like to prove as exercise.

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Let $V$ be a vector space with $dim_{\mathbb{K}}(V)=n \in \mathbb{N}$ and $T\in End(V)$. The following facts are equivalent:

1. $V$ is $T$-indecomposable;
2. $V$ is $T$-cyclic and $m^T(t)=p(t)^k$, where $p(t)$ is a suitable irreducible and monic polynomial in $\mathbb{K}[t]$ and $k\geq 1$.

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I add some notations and terminologies.

• $m^T(t)$ is the minimal polinomial of $T$.

• A vector space is $T$-invariant if $T(V)\subseteq V$.

• A vector space is $T$-cyclic if there exists a $u\in V$ such that $$ V\ =\ <u>_T\ =\ \{p(T)(u):p(t) \in \mathbb{K}[t]\}$$

• We say that a $T$-invariant vector space $V$ is $T$-indecomposable if there do not exist non-trivial $T$-invariant subspaces $U$ and $W$ such that $V=U\oplus W$.

Theorem of primary decomposition of a vector space

Let $V$ be a vector space with $dim_{\mathbb{K}}(V)=n \in \mathbb{N}$ and $T\in End(V)$. We suppose that $m^T(t)=\prod_{k=1}^rp_k(t)^{e_k}$. Then:

1. $V= ker(p_1(T)^{e_1})\oplus\dots\oplus ker(p_r(T)^{e_r})$
2. $C^T(t)=\prod_{j=1}^rp_j(t)^{m_j}$, where $m_j\geq e_j$
3. $ker(p_j(T)^{e_j})=ker(p_1(T)^{m_j})$
4. $\dim ker(p_j(T)^{e_j})=m_j\deg p_j$
5. Set $T_j$ the restriction of $T$ on $ker(p_j(T)^{e_j})$, then $m^{T_j}(t)=p_j(t)^{e_j}$ and $C^{T_j}(t)=p_j(t)^{m_j}$.

Theorem of primary decomposition of a $T$-cyclic vector space

Let $V$ be a vector space with $dim_{\mathbb{K}}(V)=n \in \mathbb{N}$ and $T\in End(V)$. We suppose that $m^T(t)=p(t)^{e}$. Then:

1. $V= U_1\oplus\dots\oplus U_h$, where $U_i$ is $T$-cyclic subspace of $V$.
2. Set $T_i$ the restriction of $T$ on $U_i$, then $m^{T_i}(t)=p(t)^{e_i}$, where $e=e_1\geq e_2\geq \dots\geq e_p\geq 1$.
3. $\dim U_i=e_i\deg q(t)$.
4. This decomposition is unique, unless the order of subspaces.

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I try to summarize my proof.
$(1)$ implies $(2)$. Since $V$ is $T$-indecomposable, then $m^T(t)$ is a power of an irreducible and monic polynomial in $\mathbb{K}[t]$. In fact, by contraddiction we suppose that $m^T(t)=p_1(t)^{k_1}p_2(t)^{k_2}$, where $p_1(t)$ and $p_2(t)$ are irreducible and monic polynomials in $\mathbb{K}[t]$. Using the Theorem of primary decomposition of a vector space, we have $$ V= ker(p_1(T)^{k_1})\oplus ker(p_2(T)^{k_2})$$ hence $V$ is a direct sum of two non-trivial $T$-invariant subspaces of $V$, so a contraddiction. Therefore $m^T(t)=p(t)^k$, where $p(t)$ is a suitable irreducible and monic polynomial in $\mathbb{K}[t]$ and $k\geq 1$. Now, using the Theorem of primary decoposition of $T$-cyclic vector space, we have that $V$ is $T$-cyclic.
$(2)$ implies $(1)$. We suppose by contraddiction that $V$ is not $T$-indecomposable, so $V=U\oplus W$, where $U$ and $W$ are two non-trivial $T$-invariant subspaces of $V$. We set $T_{|U}=f$ and $T_{|W}=g$. Then $m^T(t)=m^f(t)m^g(t)=p(t)^k$. Hence $m^f(t)=p(t)^r$ and $m^f(t)=p(t)^s$, with $r+s=k$. Using the Theorem of primary decoposition of $T$-cyclic vector space, we have that $$ U=U_1\oplus\dots\oplus U_p $$ where $U_i$ is a $T$-cyclic subspaces of $V$ for $i=1,\dots,p$ and $m^{f_{|U_i}}(t)=p(t)^{z_i}$ with $z_1=r\geq \dots z_p>0$. The same holds for $W$. Then we have that $V$ is a direct sum of $T$-cyclic subspaces such that the minimal polynomial of a restriction of $T$ on a such subspace is a power of $p(t)$. Since $V$ is $T$-cyclic for hypotesis, then we a contraddiction with the uniqueness of $T$-cyclic decomposition.

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If you disagree with me, can you post some suggestions to improve the proof please?
Moreover, we have some other characterizations of $T$-indecompasable spaces in terms of other things?