= Solution
Every $h\in G$ permutes the factors:
$$
h\bigl(f(t,\alpha)\bigr)
=\prod_{g\in G}(t-hg(\alpha))
=f(t,\alpha).
$$
Thus all coefficients lie in the <fixed field> $L^G$. Let $E$ be the field they generate over $K$. Then
$$
K\subseteq E\subseteq L^G.
$$
Because $L=K(\alpha)=E(\alpha)$ and $f(t,\alpha)\in E[t]$ has degree $|G|$ with root $\alpha$,
$$
[L:E]\leq|G|.
$$
Part (a) and the <tower law for field extensions> give
$$
[L:E]=[L:L^G][L^G:E]=|G|[L^G:E].
$$
Consequently $[L^G:E]=1$, so
$$
\boxed{E=L^G.}
$$
This is the <coefficients of an orbit polynomial for a primitive generator> result.
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