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.
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Let $L,M$ be the splitting field of $f,g$ over $K$. Then the compositum $LM$ is the splitting field of $fg$. Now your question translates with the help of the main theorem of galois theory into the following group question: How can one compute $G$ if one knows normal subgroups $N_1, N_2$, the quotients $G/N_i$ and the fact that $G=N_1 N_2$. This seems pretty hard to me in general. 


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. 

