Squarefree polynomials with prescribed coefficients

For nonempty subsets $S_0,\dots ,S_{n-1}$ of a (large enough) finite field $\mathbb{F}$ satisfying $$\mathrm{\mid }{S}_1\mathrm{\mid },\dots ,\mathrm{\mid }{S}_{n-1}\mathrm{\mid }>2\mathrm{o}\mathrm{r}\mathrm{\mid }{S}_1\mathrm{\mid },\mathrm{\mid }{S}_{n-1}\mathrm{\mid }>n-1,$$ we show that there exist $a_0 \in S_0, \dots, a_{n-1} \in S_{n-1}$suchthat$$ T^n + a_{n-1}T^{n-1} + \dots + a_1T + a_0 \in $\mathbb{F}$[T] $$ is a squarefree polynomial.