This is an exercise about Galois theory on page595 in Dummit&Foote's Abstract Algebra.
Let $f (x) \in F[x]$ be an irreducible polynomial of degree $n$ over the field $F$, let $L$ be the splitting field of $f(x)$ over $F$ and let $\alpha$ be a root of $f(x)$ in $L$. If $K$ is any Galois extension of $F$, show that the polynomial $f(x)$ splits into a product of $m$ irreducible polynomials each of degree $d$ over $K$, where $d = [K(\alpha) : K] = [(L \cap K)(\alpha) : L \cap K]$ and $m = n /d = [F(\alpha) \cap K : F]$. [Show first that the factorization of $f(x)$ over $K$ is the same as its factorization over $L \cap K$. Then if $H$ is the subgroup of the Galois group of $L$ over $F$ corresponding to $L \cap K$ the factors of $f (x)$ over $L \cap K$ correspond to the orbits of $H$ on the roots of $f (x)$. Use Exercise 9 of Section 4.1.]
The hint says that "the Galois group of $L$ over $F$", but $L$ is not necessarily Galois since $f(x)$ is not necessarily separable, so where need to be corrected about this question? [Additionally suppose that $L/F$ is Galois (i.e., $f(x)$ is separable), or suppose $\text{char}\,F=0$(so that $F$ is a perfect field for which irreducibility implies separability), or something else?]
There are some related links:
Determining the number and degree of irreducible factors over Galois extension
About Galois extensions contained in inseparable extensions
Update:
Exercise 9 of Section 4.1 says that if $G$ acts transitively on a finite set $A$ and $H\unlhd G$, let $\mathcal{O}_1,\cdots,\mathcal{O}_r$ be the distinct orbits of $H$ on $A$, then for a fixed $a\in A$ we have for each orbit $\mathcal{O}_i=\mathcal{O}_{a_i}$, $$ \begin{cases}|\mathcal{O}_i|=|A|/r\\|H\cap G_a|=|G|/|\mathcal{O}_i|\\|HG_a|=|G|/r\end{cases} $$ where $G_a$ is the stabilizer of $a\in\mathcal{O}_i$ in $G$.
If $f(x)$ is separable then $L/F$ is Galois so that the intersection $L\cap K$ of two Galois extensions is also Galois and hence $H=\mathbf{Gal}(L/L\cap K)\unlhd\mathbf{Gal}(L/F)=G$ by the Fundamental Theorem of Galois theory, then regarding $A$ as the set of roots of $f(x)$ we can use Exercise 9 of Section 4.1.
But if not, how to go forward?
