Galois group of an irreducible cubic

An irreducible cubic over a field of characteristic other than two has Galois group $A_3$ when its discriminant is a square and $S_3$ otherwise.

If a separable cubic has polynomial discriminant $\Delta$, then adjoining $\sqrt{\Delta }$ gives the subfield fixed by the alternating part of its Galois group. When the cubic Galois group is $S_3$, this is its unique quadratic subfield.

The polynomial $X^3-X-1$ is irreducible over $\mathbb{Q}$ and has discriminant $-23$, so its splitting field has Galois group $S_3$. Its unique quadratic subfield is the fixed field of $A_3$, namely $\mathbb{Q}(\sqrt{-23})$.

The discriminant square class detects whether the transitive cubic Galois group lies in the alternating group.