Let $K$ be a field of characteristic 0 , and let $P(X)=X^{4}+b X^{2}+c X+d$ be an irreducible quartic polynomial over $K$. Let $\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}$ be its roots in an algebraic closure of $K$, and consider the Galois group $\operatorname{Gal}(P)$ (the group $\operatorname{Gal}(F / K)$ for a splitting field $F$ of $P$ over $K$ ) as a subgroup of $S_{4}$ (the group of permutations of $\left.\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right)$.

Suppose that $\operatorname{Gal}(P)$ contains $V_{4}=\{1,(12)(34),(13)(24),(14)(23)\}$.

(i) List all possible $\operatorname{Gal}(P)$ up to isomorphism. [Hint: there are 4 cases, with orders 4 , 8,12 and 24.]

(ii) Let $Q(X)$ be the resolvent cubic of $P$, i.e. a cubic in $K[X]$ whose roots are $-\left(\alpha_{1}+\alpha_{2}\right)\left(\alpha_{3}+\alpha_{4}\right),-\left(\alpha_{1}+\alpha_{3}\right)\left(\alpha_{2}+\alpha_{4}\right)$ and $-\left(\alpha_{1}+\alpha_{4}\right)\left(\alpha_{2}+\alpha_{3}\right)$. Construct a natural surjection $\operatorname{Gal}(P) \rightarrow \operatorname{Gal}(Q)$, and find $\operatorname{Gal}(Q)$ in each of the four cases found in (i).

(iii) Let $\Delta \in K$ be the discriminant of $Q$. Give a criterion to determine $\operatorname{Gal}(P)$ in terms of $\Delta$ and the factorisation of $Q$ in $K[X]$.

(iv) Give a specific example of $P$ where $\operatorname{Gal}(P)$ is abelian.