Suppose $\displaystyle f(x) $ is relatively prime to $\displaystyle 0_F $. Then $\displaystyle \gcd(f(x), 0_F) = 1_F $. So $\displaystyle 1_F = f(x)u(x)+0_{F}v(x) $ for some polynomials $\displaystyle u(x) $ and $\displaystyle v(x) $. Thus $\displaystyle 1_F = f(x)u(x) $.

So this means that $\displaystyle f(x) $ is a unit?