A Worked out Galois Group for the Classroom

Let f = X6 − 3X2 − 1 ∈ Q[X] 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.