Let $V$ be a finite dimensional vector space over a field $F$ and let $T: V\to V$ be a linear transformation. Assume that $T(W)$ is not a subspace of $W$ for every non-zero, proper subspace $W$ of $V$. Let $F[T]:=\{ p(T): V\to V | p(x)\in F[x]\}$.
I have three questions :
(1) How to show that $\ker p(T)=0$ for every $0\ne p(T)\in F[T]$ ?
(2) if (1) is true then every $0 \neq p(T)\in F[T]$ is an isomorphism. How to show that the inverse of every $0 \neq p(T)\in F[T]$ is again a polynomial in $T$ ? i.e. how to show that $F[T]$ is a field ?
(3) How to show that $[F[T] : F]=\dim_F (V)$ ? i.e. how to show that $\dim_F F[T]=\dim_F V$ ?
For (1), it suffices to note that $\ker p(T)$ is an invariant subspace of $T$ (for any polynomial $p$).
For (2): Let $m(x)$ be the minimal polynomial of $T$. Using condition (1), we conclude that $m(x)$ is irreducible (how?). Consider an arbitrary polynomial $p(x)$. Since the gcd of $p$ of $m$ is $1$, there exist polynomials $f,g$ such that $$ f(x) p(x) + g(x)m(x) = 1. $$ Because $m(T) = 0$, we can deduce that $f(T)$ is the inverse of $p(T)$.
For (3): Note that $[F[T]:F]$ is the degree of $m$. By the Cayley-Hamilton theorem, this is at most $\dim V$. To show that it cannot be less than $\dim(V)$, note that for any non-zero $x \in V$, the $T$-invariant subspace generated by $x$ is a non-zero subspace of $V$ whose dimension is at most $\deg(m)$.