A note on freeness

In this brief note we show that for a field extension $K\mathrm{/}F$, $S=K[\text{\hspace{0.17em}⁣}[\mathbf{x}]\text{\hspace{0.17em}⁣}]$ is a free $R=F[\text{\hspace{0.17em}⁣}[\mathbf{x}]\text{\hspace{0.17em}⁣}]$-module precisely when $K\mathrm{/}F$ is finite. We then raise the question \emph{what is the projective dimension of $S$?}