I am trying to show that and are not isomorphic.

My question is this... Isomorphisms preserve subfields (including, in this case, ), and I WANT to say something like ' must be preserved so the mapping acts as the identity on but then it must act as the identity on the whole field, but then clearly it is not an isomorphism so and cannot be isomorphic.'

But, is any of this true, or even a useful way of thinking about it? Sadly, I suspect not...