Example of a normal extension. Can you give an example of a Normal extension which is not a splitting field of some polynomial.? I know that splitting field of a polynomial is always a normal extension but i am looking for the converse. I am sure that the normal extension will have to be infinite.

I guess it depends on how one defines "normal" for infinite extensions. Let $$K={\bf Q}(\sqrt2,\sqrt3,\sqrt5,\sqrt7,\dots)$$ Then $K$ is normal over the rationals, in the sense that any polynomial irreducible over the rationals with a zero in $K$ splits into linear factors over $K$. But it's not a splitting field, in that there is no single polynomial $f$ such that $K$ is obtained from the rationals by adjoining the roots of $f$. 
