I am working on the following question:
Let $F$ be a field. Let $F^n$ be an $F[x]$ module where $x$ acts on $(a_1, ..., a_n) \in F^n$ by $x \cdot (a_1, \dots, a_n) = (0, a_1, \dots, a_{n - 1})$. Prove $F[x]/(x^n) \cong F^n$ as $F[x]$-modules. Use this isomorphism to describe the $F[x]$-submodules of $F^n$.
My idea so far:
Let $\varphi\colon F[x] \to F^n$ be given by $\varphi(p(x)) = \varphi(\sum_{i = 1}^m a_i x^{i - 1}) = \sum_{i = 1}^m x^{i - 1} (a_i, 0, \dots, 0)$. I believe that $\varphi$ is a surjective $F[x]$-module homomorphism with $\ker\varphi = (x^n)$. So by the first isomorphism theorem, $F[x]/(x^n) \cong F^n$.
However, I am unsure how to use this isomorphism to describe the $F[x]$-submodules of $F^n$.