I am trying to solve the problem:

Let $V$ be a vector space and $T$ a linear transformation $T:V \to V$. Let $(V,T)$ be a $K[x]$-module. 

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> Show that $(V,T)$ is simple if and only if $V$ is finite dimensional and $\mathcal X\_T=m\_T$ is irreducible (where $\mathcal X\_T$ is the characteristic polynomial and $m\_T$ is the minimal).
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I am pretty lost with the exercise, I would appreciate any suggestions on how could I show both implications.