What automorphism cannot be extended when the extension is not normal?

Question

As I understand it, if we have an automorphism $\phi : K \rightarrow K$, and a finite normal extension $N/K$, $\phi$ can always be extended to an automorphism of $N$. But what happens if $N/K$ is not a normal extension? If for instance I consider $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$, what automorphism $\phi$ could not be extended?

Answer 1

Consider the field $K=\Bbb Q(\sqrt2\,)$ with its nontrivial automorphism $\phi:\sqrt2\mapsto-\sqrt2$. Consider also the quadratic (and thus normal) extension $\Bbb Q(\sqrt[4]2\,)=N\supset K=\Bbb Q(\sqrt2\,)$. As you see, there is no extension of $\phi$ to $N$. So your premise is false.

I’ve been considering writing one last paper, an examination of this problem in more generality.