Let $F$ be a field and $f(x)$ in $F[x]$ satisfy $f(x)=f_1(x)f_2(x)\cdots f_n(x)$, where $f_i(x)$ is irreducible in $F[x]$.
My opinion is:
If $E$ is splitting field of $f(x)$ and we denote degree of $f_i(x)$ by $d_i$, then $\mathrm{Gal}(E/F) \leq S_{d_1} \times S_{d_2} \times S_{d_3} \times \cdots \times S_{d_n} $ ($\mathrm{Gal}(E/F)$ is Galois group of $f(x)$ and $S_{n}$ is permutation group of $n$)
My Question: Is this collect?
If not, is there similar theorem of this opinion? what condition do I need?
For example, if $f(x)$ is separable, then this is true?