Lemma 5

Let $X$ be a finite $\Gamma$-set and let $x_1, \ldots, x_n$ be points of $X$ such that the orbits $\Gamma x_1, \ldots, \Gamma x_n$ form a partition of $X$. For $1 \leq i \leq n$ let $\Delta_i$ be the stabilizer of $x_i$ in $\Gamma$ and let $L_i$ be the field of invariants of $A_i$. Then $L_1, \ldots, L_n$ are separable extensions of finite degree of $K$, and the mapping $f \mapsto (f(x_1), \ldots, f(x_n))$ is a K-algebra isomorphism of $\Theta(X)$ onto $L_1 \times \ldots \times L_n$.

By hypothesis the subgroups $A_i, \ldots, A_n$ of $\Gamma$ are open and Cor. 5 of V, p. 68, shows that the subextensions $L_1, \ldots, L_n$ of $K$, are of finite degree over $K$. Clearly they are separable; now the last assertion of Lemma 5 is immediate.

From Lemma 5 and Th. 4 (V, p. 34f.) we obtain immediately the following result.