I'm trying to prove that splitting fields are unique up to isomorphism.
Say is a splitting field for a polynomial over a field . Suppose also that is an isomorphic extension, ie that we have and isomorphisms with .
How can I show that is a splitting field of over ?
I feel this should be intuitively obvious and that it should fall easily out of definitions and very basic facts, yet I can't seem to be able to prove it nicely.