 Coefficients of an orbit polynomial for a primitive generator

If $K\subseteq L^G$ and $L=K(\alpha )$, then the coefficients of $f(t,\alpha )$ generate $L^G$ over $K$. Indeed, for the field $E$ generated by those coefficients, $L=E(\alpha )$ and $[L:E]\le |G|$, while the Artin fixed-field theorem gives $[L:L^G]=|G|$.