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It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedean field $K$ has solvable (in fact supersolvable [edit: no!]) Galois group $G$. One sees this by using the ramification ...

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.