This came up in a thread a couple of days ago, I believe. Take and look at where maps to. Square it, and then in a few lines you can see that the thing you're mapping to must be outwith .
What is it you find unsatisfactory about your proof?
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
Thanks I have something similar. I guess the reason I feel unsatified is the proof is out of place in my text. In Dummit and Foote this is in the beginning of chapter 14(the beginning of Galois theory), but I didn't use anything from the current chapter. I used only tools from the previous chapter 13. Of course this is not a reason to think a proof is incorrect, but it does make me suspect that I may have missed an applicaiton of a more recent idea.