I proved this simple thing, but using some simple field theory. I want to know whether I can prove it with simpler tools. The proof is not difficult, it uses only a little field theory, like the idea of extensions of finite degree. But can I prove it only using the definitions?
Another related question: If I know that the element $u$ satisfies some polynomial equation $p(x)=0$, can I find the polynomial for $1/u$ only knowing $p(x)$?