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:

(i)

and

(ii)

satisfying

(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.