Let $K$ be a field of characteristic $\neq 2,3$, and assume that $K$ contains a primitive cubic root of unity $\zeta$. Let $P \in K[X]$ be an irreducible cubic polynomial, and let $\alpha, \beta, \gamma$ be its roots in the splitting field $F$ of $P$ over $K$. Recall that the Lagrange resolvent $x$ of $P$ is defined as $x=\alpha+\zeta \beta+\zeta^{2} \gamma$.

(i) List the possibilities for the group $\operatorname{Gal}(F / K)$, and write out the set $\{\sigma(x) \mid \sigma \in \operatorname{Gal}(F / K)\}$ in each case.

(ii) Let $y=\alpha+\zeta \gamma+\zeta^{2} \beta$. Explain why $x^{3}, y^{3}$ must be roots of a quadratic polynomial in $K[X]$. Compute this polynomial for $P=X^{3}+b X+c$, and deduce the criterion to identify $\operatorname{Gal}(F / K)$ through the element $-4 b^{3}-27 c^{2}$ of $K$.