Like I mentioned in the question, I am trying to find a way to find the Galois group for a general quartic polynomial. I am reading the book Galois Theory of Escofier. He solved the case of cubic polynomials nicely using resolvents. But it is not clear how to use it for quartic polynomials.
Can you recommend me some recourses on how to compute Galois groups of Quartic polynomal? Any methods would be great, not necessarily on resolvents.
There is a very nice text about Galois groups of cubic and quartic polynomials by Keith Conrad. The possible Galois groups are exactly $S_4,A_4,D_4,V_4$ and $C_4$, see section $3$ and $4$. With this it is easy to decide for a given quartic polynomial which Galois group it has.