Let $N/K$ be a normal extension of fields. Let $f\in K[X]$ be an irreducible polynomial with monic irreducible factors $g,h\in N[X]$. Show that there exists an automorphism $\varphi$ on $N$ which induces an automorphism $\Phi$ on $N[X]$ (via $\varphi$ on coefficients of a polynomial) such that $\Phi(g)=h$. Determine whether the result holds true if $N/K$ is not necessarily normal.
I think it does not hold for general extension $N/K$. The result shows that $g$ and $h$ has the same degree. So if we consider $\mathbb{Q(\sqrt[3]{2})}/\mathbb{Q}$ and $f=X^3-2=(X-\sqrt[3]{2})(X^2+X+\sqrt[3]{2}^2)$. Let $\alpha=\sqrt[3]{2}\zeta$. Then the minimal polynomial of $\sqrt[3]{2}$ is $X-\sqrt[3]{2}$ and has degree $1$ while the minimal polynomial of $\alpha$ is $X^2+X+\sqrt[3]{2}^2$ and has degree $2$. So there is no such $\varphi$. Am I right?
I do not know how to prove the result. Any ideas?