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...