Let be a separable cubic with Galois group S 3 and roots u 1 ,u 2 ,U 3 F Then the distinct intermediate fields of the extension of K by F are F, K(), K(u 1 ), K(u 2 ), K(u 3 ), K The corresponding subgroups of the Galois group are 1 ,A 3 , T 1 , T 2 , T 3 and S 3 where T i (1),(jk) j i k ...

We can use the fact that the Galois group of a separable polynomial of degree 3 is isomorphic to S3 ...

Let be a separable cubic with Galois group S 3 and roots u 1 ,u 2

Question:

Let ∫ be a separable cubic with Galois group S₃ and roots u₁,u₂,U₃ ϵ F. Then the distinct intermediate fields of the extension of K by F are F, K(Δ), K(u₁), K(u₂), K(u₃), K. The corresponding subgroups of the Galois group are 1 ,A₃, T₁, T₂ , T₃ and S₃ where T_i = {(1),(jk) |j ≠ i ≠ k}