I am working on finding the Galois groups of polynomials - in particular polynomials of degree $4$
I know that if we have a polynomial of the form $X^4+pX^2+qX+r$ we can find its cubic resolvent. That is: $$U^3+2pU^2+(p^2-4r)U-q^2$$
How can we find the discriminant of this resolvent cubic?
I know that if we know the discriminant (in particular if it is square or not) and the reducibility of the resolvent cubic then we can find its Galois group by distinguishing between $A_4, V_4, S_4, C_4$ etc
Thank you