What are the possible Galois groups for a cubic polynomial \(f(x) \in K[x]\) over a field \(K\) (of characteristic \(\ne 2, 3\))? Classify them completely using the discriminant and irreducibility.
Solution
Let \(L\) be the splitting field of a cubic \(f\in K[x]\), with \(\operatorname{char}K\ne2,3\).
If \(f\) splits over \(K\), then \(L=K\) and the Galois group is trivial. If \[ f=(x-a)q(x) \] with \(q\) irreducible quadratic, then \(L\) is the quadratic splitting field of \(q\), so \[ \operatorname{Gal}(L/K)\cong C_2. \]
Now suppose \(f\) is irreducible. Then its roots are distinct and the Galois group \(G\le S_3\) acts transitively, so \[ G\cong A_3\quad\text{or}\quad S_3. \] Let \[ \delta=(\alpha_1-\alpha_2)(\alpha_1-\alpha_3)(\alpha_2-\alpha_3), \qquad \Delta(f)=\delta^2. \] A permutation of the roots sends \(\delta\) to its sign times \(\delta\). Therefore \[ G\subseteq A_3 \iff \delta\in K \iff \Delta(f)\in K^{\times2}. \] Consequently, \[ \operatorname{Gal}(L/K)\cong \begin{cases} C_3,& f\text{ irreducible and }\Delta(f)\text{ is a square in }K,\\ S_3,& f\text{ irreducible and }\Delta(f)\text{ is not a square in }K. \end{cases} \] These, together with the trivial and \(C_2\) reducible cases, are all possibilities.