I am taking an introductory number theory course this term, and I have found that while my algebra has been for the most part sufficient, I am severely lacking in even the basics concerning fields and field extensions.
My question is, what are the basic concepts/fundamental theorems in the theory of fields and field extensions? For group theory, for instance, I would say that the first isomorphism theorem, the class equation, the Sylow theorems, and a healthy familiarity with the likes of $Z/nZ$ and $S_n$ encapsulate the top parts of what one might expect to learn in a one semester introduction to algebra (by which I don't mean that this is all that one would learn, but that a lot of the other things done would be done to support these top level concepts). Is there a similar list of concepts which would similarly encapsulate some sort of top level knowledge of the subject of fields and field extensions? Any good introductions to the subject are also much appreciated!