Let $K$ be a algebraically closed field with $char(K) = p$. For integer $r>0$ and $c_0, c_r \neq0$, let $f(X)$ be a $K$-coefficient polynomial with $p^r$ degree,
$f(X)=c_0X+c_1X^p+...+c_rX^{p^r} $.
We let denote $f^k(X)$ composition of $f$ with k-times.
Let $A$ be a $F_p[T]$ ($F_p$ is a prime field). We define action of $A$ to $K$ as follows. For $a= \sum_{k=0}^{n} a_k T^k \in A$ and $x \in K$,
$ax = \sum_{k=0}^{n} a_k f^k(x)$.
From this action, $K$ is a $A$-module. We let denote $V_a= \{ x \in K|ax=0 \}$ for $a \in A$.
(1) If $a\neq 0$, then $V_a$ is finitely generated $A$-module.
(2) Suppose $a = T$. Find structure of $V_a$ as $A$-module with following format
$V_a \simeq A^{\bigoplus m} \bigoplus (A/a_1A) \bigoplus (A/a_2A) \bigoplus ... \bigoplus (A/a_lA)$.
(this means get $a_i$ explicitely)
(3) Show $K$ is not finitely generated $A$-module.
My approach
(1) $ax$ is a polynomial with $(p^r)^n$ degree. Since $K$ is algebraically closed, the number of element of $V_a$ is at most $(p^r)^n$. Hence finitely generated $A$-module.
(2) From structure theorem on PID (https://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain), we can describe $V_a$ as above. However, I cannot determine each $a_i$.
(3) Maybe consider cover of $K$ by several $V_a$ and get contradiction somehow.
Thank you.