Why doesn't $f_1, f_2$ have an inverse polynomial?

Question

Consider $f\in R=F[X]$. It is given that $f$ doesn't have an inverse but it's reducible. Therefore, there are $f_1,f_2$ such that $f=f_1f_2$, where $f_1, f_2$ also doesn't have an inverse polynomial.

I can see why it's true that they both can't have an inverse. But why isn't it possible for $f_1$ (WLOG) to have an inverse?

Thanks.

Answer 1

Is $F$ a field? Then the only polynomials that have multiplicative inverses are constants.