I need to prove that

So here is my thoughts on this:

Since

is the prime subfield is must be fixed by any isomorphism from

Since

is a basis for

and

is a basis for

The isomorphism must be of the form

or

Both of these only work for the addative group structure, but don't preserve multiplication. So they are not isomorphisms.

Therefore

I feel unsatisfied with the above proof. Another thought that I had was since \mathbb{Q}(\sqrt{2}) has only two automorphism showing that neither of those could be made into an isomorphism to

but that came up empty.

Any thoughts or comments would be awsome.

Thanks

TES