Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 18H / a / Solution

For roots $r_1,r_2,r_3$, the discriminant isand for $f=X^3+pX+q$,If $f$ is irreducible, its Galois group is transitive in $S_3$: it is $A_3\cong C_3$ when $\Delta$ is a square in $K$, and $S_3$ otherwise. If $f$ has one root in $K$ and an irreducible quadratic factor, the group is $C_2$; if it splits, it is trivial. A repeated-root depressed cubic already has all roots in $K$, so adds no case.

For $X^3+3X+1$, the rational-root test proves irreducibility andis not a rational square. Thus $G\cong S_3$. By the fundamental theorem of Galois theory, the complete subfield list iswhere the three cubic fields fix the three order-two subgroups and the quadratic field fixes $A_3$.