Exercise 16, § 10

Let a be an integer and let x₁, x₂, x₃, x₄ be the roots of P(X) = X⁴ − aX − 1 in an algebraic closure of Q. We identify the Galois group G of P with a group of permutations of {x₁, ..., x₄}. Suppose that Q(x₁) contains no quadratic subextension (cf. Ex. 1 of V, p. 154). a) Show that G = S₄ or A₄. (Use Ex. 1 of V, p. 154: P is irreducible, hence G acts transitively; further Q(x₁) contains no quadratic subextensions, hence the stabilizer H of x₁ is not contained in a subgroup of index 2 of G, whereas [G : H] = 4; deduce that G is not a 2-group, hence its order is divisible by 3.) b) Let 6 = $\prod_{1 \leq i < j \leq 4} (x_i - x_j)$ and d = δ². Show that d = 4⁴ − 3³ a⁴; further s(δ) = εₛδ for all s ∈ G. Deduce that G = A₄ if δ ∈ Q, and G = G, otherwise. Show that 6 is not rational. (We have thus constructed an infinite family of equations of degree 4 over Q with Galois group S₄.)