TITLE: Galois group of a product of polynomials
QUESTION [13 upvotes]: 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.

REPLY [3 votes]: Here you can find the proof of $Gal(E_1 E_2) = Gal(E_1) \times_{Gal(E_1 \cap E_2)} Gal(E_2)$ and an example how this can be used to compute the Galois group of a product of two polynomials.