Splitting field of x cubed minus x minus one
= Splitting field of x cubed minus x minus one
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 $\mathbb Q(\sqrt{-23})$.