Let $N/F$ be a normal extension, let $X$ be an indeterminate, let $f\in F[X]$ be an irreducible polynomial, and let $g_1,g_2\in N[X]$ be irreducible monic factors of $f$.
We want to find an $F$-automorphism of $N$ which maps $g_1$ to $g_2$.
Let $N^a$ be an algebraic closure of $N$, and $\alpha_i$ a root of $g_i$ in $N^a$.
The minimal polynomial of $\alpha_i$ over $F$ being $f$, there is an $F$-automorphism $\sigma$ of $N^a$ mapping $\alpha_1$ to $\alpha_2$.
As $N/F$ is normal, we have $\sigma N=N$.
The minimal polynomial of $\alpha_i$ over $N$ being $g_i$, our automorphism $\sigma$ maps $g_1$ to $g_2$.