Proposition 22

Let $A$ be a (commutative) $K$-algebra and $G$ a locally finite group of operators on $A$. Then $A$ is integral over the subalgebra $A^G$.

For all $x \in A$, let $x_i$ ($1 \leq i \leq n$) be the distinct elements of the orbit of $x$ under $G$; for all $\sigma \in G$, there exists a permutation $\pi_\sigma$ of the set $\{1, 2, \ldots, n\}$ such that $\sigma.x_i = x_{\pi_\sigma(i)}$ for $1 \leq i \leq n$; therefore the elementary symmetric functions of the $x_i$ are elements of $A$ which are invariant under $G$, in other words elements of $A^G$. As $x$ is a root of the monic polynomial $\prod_{i=1}^n (X - x_i)$ and the coefficients of this polynomial belong to $A^G$, $x$ is integral over $A^G$.