Decomposition of Galois group

Question

Let $E$ be the splitting field a (not necessarily irreducible) polynomial $f$ over $F$. Say $f=gh$ with $g,h$ being coprime. Let $F_g$ and $F_h$ be the splitting fields of $g$ and $h$ respectively. Is it true that $$\text{Gal}(E/F)\cong \text{Gal}(F_g/F)\times \text{Gal}(F_h/F)$$

Answer 1

Take any two coprime polynomials with the same splitting field for a counter-example.

Examples:

• $x^2+1, x^2+4 \in \Bbb Q[X]$
• $x^p-t, x^p-t-1 \in \Bbb F_p(t)[X]$