I was hoping that someone familiar with a bit of commutative algebra could help me see why this is true.

Suppose that $\displaystyle R,S$ are commutative rings with identity, such that $\displaystyle S$ is faithful as an $\displaystyle R$-algebra (i.e., there is a monomorphism $\displaystyle \theta :R\rightarrow S$). In this way, we may identify $\displaystyle R$ with $\displaystyle \theta (R)$ and view $\displaystyle R\subseteq S$.

Let $\displaystyle G\leq \mathrm{Aut}_R(S)$ and say $\displaystyle |G|< \infty$ (that is, $\displaystyle G$ is a finite group of ring automorphisms of $\displaystyle S$ such that elements of $\displaystyle R$ are fixed pointwise by these maps).

Define $\displaystyle S^G=\{x\in S\mid \sigma (x)=x\,\forall\, \sigma\in G\}$. We say that $\displaystyle S$ is a

__normal__ extension of $\displaystyle R$ with group $\displaystyle G$ if $\displaystyle S^G=R$ (clearly we have the containment $\displaystyle R\subseteq S^G$ by definition of $\displaystyle G$).

We say that $\displaystyle S$ is a

__Galois__ extension of $\displaystyle R$ if:

(i) $\displaystyle S^G=R$ and

(ii) $\displaystyle \exists\, x_1,\ldots ,x_n,y_1,\ldots y_n\in S$ satisfying

$\displaystyle \sum_{i=1}^n x_i\sigma(y_i)=\delta_{\sigma,1}$

(where $\displaystyle \delta$ is the Kronecker delta, and $\displaystyle 1\in G$ represents the identity automorphism of $\displaystyle S$).

The theorem is that, if $\displaystyle S$ is a Galois extension of $\displaystyle R$ with Galois group $\displaystyle G=\mathrm{Gal}_R(S)$, then we may view $\displaystyle S\otimes _RS$ as a Galois extension of $\displaystyle R$ (identifying $\displaystyle R$ with $\displaystyle R\otimes _R1\subseteq S\otimes _RS$ with Galois group $\displaystyle G\times G$. The group $\displaystyle G\times G$ acts on the algebra $\displaystyle S\otimes _RS$ via the action

$\displaystyle (\sigma,\tau)[\sum x_i\otimes y_i]=\sum \sigma(x_i)\otimes \tau(y_i)$.

Right now, I'm just trying to show that $\displaystyle S\otimes _RS$ is a normal extension of $\displaystyle R$ (that is, showing $\displaystyle (S\otimes _RS)^{G\times G}=R$). The containment $\displaystyle R\subseteq (S\otimes _RS)^{G\times G}$ is clear, but because of how complicated expressions in $\displaystyle S\otimes _RS$ can be (as full summations of simple tensors), I can't see why the other containment must be true.

Anyone have any ideas at all? I could really use some help trying to understand this.