A Worked out Galois Group for the Classroom

Let f=X6−3X2−1∈QX and let Lf be the splitting field of f over Q. We show by hand that the Galois group Gal(Lf/Q) of the Galois extension Lf/Q is isomorphic to the alternating group A4. Moreover, we show that the six roots of f correspond to the six edges of a tetrahedron and that the four roots of the polynomial X4+18X2−72X+81 correspond to the four faces of a tetrahedron, which allows us to determine all eight proper intermediate fields of the extension Lf/Q.