3. Let f(x) = Q[x] be an irreducible cubic with three complex roots a₁, a2, a3. Let D = (a₂-a3)(a3 - a₁). Let G be the G

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3 Let F X Q X Be An Irreducible Cubic With Three Complex Roots A A2 A3 Let D A A3 A3 A Let G Be The G 1 (113.44 KiB) Viewed 34 times

3. Let f(x) = Q[x] be an irreducible cubic with three complex roots a₁, a2, a3. Let D = (a₂-a3)(a3 - a₁). Let G be the Galois group of f, thought of as a subgroup (a₁ - α₂) of S3. (a) Show that o(D) = sgn(o)D. for all o E G. (Here sgn is the sign of a permutation). (b) Show that G is a subgroup of the alternating group A3 if and only if D E Q. (c) Generalise this to give a criterion for when a Galois group is a subgroup of An for any n.