A Worked out Galois Group for the Classroom

Abstract

Let f = X 6 − 3 X 2 − 1 ∈ Q [ X ] and let L f be the splitting field of f over Q . We show by hand that the Galois group Gal ( L f / Q ) of the Galois extension L f / Q is isomorphic to the alternating group A 4. 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 X 4 + 18 X 2 − 72 X + 81 correspond to the four faces of a tetrahedron, which allows us to determine all eight proper intermediate fields of the extension L f / Q .