Orbit polynomial under a finite automorphism group

For $\alpha \in L$ and a finite automorphism group $G$, the polynomialhas coefficients in the fixed field $L^G$, because every element of $G$ permutes its factors.

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 $[L:L^G]=|G|$.

Let $L=K(\sqrt{a_1},\sqrt{a_2})$ with independent square classes and characteristic different from two, and let $G$ be generated by the automorphism negating both square roots. The two orbit polynomials are $t^2-a_1$ and $t^2-a_2$, so their coefficients generate only $K$, whereas