I'm wondering if we can apply modules structure theory to prove a standard statement about vector spaces over a field $F$: an endomorphism $\alpha : V \rightarrow V$ is diagonalisable $\iff \exists$ a polynomial $p(X)$ such that $p(\alpha) = 0$ and $p(X)$ is a product of distinct linear factors.
We would normally do this by constructing projection maps, and the proof is long, so I'm wondering if we can use $F[X]$-modules to make this proof more natural and quick.
The $\Rightarrow$ direction is okay, so let's go to the $\Leftarrow$ direction of this proof. I will strengthen the hypothesis to require the distinct linear factors to all be coprime. Let $p(X) = (X-\lambda_1)...(X-\lambda_t)$. Then by the primary decomposition the corresponding $F[X]$-module to $\alpha$ will decompose as $V_\alpha = \frac{F[X]}{(1)} \oplus ... \oplus \frac{F[X]}{(1)} \oplus \frac{F[X]}{(X-\lambda_1)} \oplus .... \oplus \frac{F[X]}{((X-\lambda_1)...(X-\lambda_t))}$. Now we want to link this to what $\alpha$ will look like and whether it will be diagonal.
How can we proceed with my method?