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.

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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 $f$.

Let $f_1,\ldots ,f_r\in K[X]$ be polynomials in the variable $X$ over the field $K$. Then the splitting field $N$ of $f_1,\ldots ,f_r$ is obtained by adjoining all of the roots of the $f_i$ to $K$. In particular $N|K$ is a finitely generated, algebraic extension and thus finite. Moreover $N$ is the splitting field of one polynomial, namely $f_1\cdot\ldots\cdot f_r$.

So the normal extension $N|K$ you are looking for must be of infinite degree. You can take the algebraic closure $\widetilde{Q}$ of the rationals $\mathbb{Q}$ for example. It is normal by definition and it is infinite because the polynomials $X^n-2$ are irreducible for every $n\in\mathbb{N}$.