If

then for all

we have that

. As

is a group we have that all the terms pair off, and so we have that

for all

.

Let

. If

then let

. It is clear that both

and

as they are invarient under automorphisms and are subfields of the ring.

So, we wish to show that

or if

,

. That is to say,

and either

or

.

Rearranging

we get that

. As

is arbitrary we have that

for all

and

. Thus,

is invariant under every single automorphism and

for all

. Thus,

or

(as

), and

, and we are done...almost.

It still needs to be shown what

is. My initial thought was that it was the prime subfield (thus, I denoted it

). However, if

is trivial this is not the case. Thus, i suspect that the prime subfield is the intersection of all the

's (it is certainly contained in the intersection). What

actually is will depend on

, and will either need someone with more knowledge of fields than I, or someone with a bigger brain...