Let $E$ be a field and let $F$ be a finite Galois extension of $E$.
Let $h(x)$ be an irreducible monic polynomial in $E[x]$, and $h_{1}(X),h_{2}(X)$ be two irreducible monic polynomials in $F[X]$, both of which divide $h(x)$.
I want to show that exists an automorphism $\theta$ of $F[X]$ such that $\theta$ leaves all elements in $E[X]$ fixed and furthermore $\theta (h_{1})=h_{2}$.
case one is h1,h2 are both linear polynomial, then its done. Suppose not, case two, I have to show that they have the same order, this is where I stuck