I encountered this problem when I was doing practice exercises on Modules over PID:
Let $F$ be a field with characteristic $\neq 2$, and let $V$ be a vector space over $F$ with basis $\left\{v_{1},v_{2},v_{3},v_{4} \right\}$. Make $V$ become an $F[x]$-module via
$$ x v_{1} = v_{2}, \quad xv_{2}= v_{1}, \quad xv_{3} =v_{4}, \quad \text{and}~ xv_{4} = -v_{3}+2v_{4}. $$ My question is: Is $V$ a free/projective $F[x]$-module? If so, how may one check it?
I understand that a module over $F[x]$ is free if and only if it is torsion free, but this seems quite hard to verify.
Any help/hint would be highly appreciated.