I was hoping that someone familiar with a bit of commutative algebra could help me see why this is true.
Suppose that are commutative rings with identity, such that is faithful as an -algebra (i.e., there is a monomorphism ). In this way, we may identify with and view .
Let and say (that is, is a finite group of ring automorphisms of such that elements of are fixed pointwise by these maps).
Define . We say that is a normal extension of with group if (clearly we have the containment by definition of ).
We say that is a Galois extension of if:
(where is the Kronecker delta, and represents the identity automorphism of ).
The theorem is that, if is a Galois extension of with Galois group , then we may view as a Galois extension of (identifying with with Galois group . The group acts on the algebra via the action
Right now, I'm just trying to show that is a normal extension of (that is, showing ). The containment is clear, but because of how complicated expressions in 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.
But since is Galois over , so in
fact and we're done...