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Let $L$ be the splitting field of $(T^2-5)(T^3-3T+1) \in \mathbb{Q}[T]$. I have to determine all subgroups of $\text{Gal}(L/\mathbb{Q})$ and all subfields $L\geq M \geq \mathbb{Q}$.

Now I asked already a similar question here and managed to do that - but in that case the subgroups were easy to determine, since my splitting field was just isomorphic to $S_3$ which I could easily show by writing down the Cayley table.

But for this exercise, I don't even know how to write down the Cayley table. My guess is, that it is isomorphic to $S_2 \times S_3$ since I have two polynomials of degree $2$ and $3$. Can someone please describe to me how to do that ? Beginning from there, I hope I should be able to do the rest on my own.

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In this case the splitting field is a composite of two spliting fields and use a result of Galois theory about the computation of the Galois group of composite in terms of the Galois groups of the two fields. – user26857 Dec 12 '12 at 10:19
In this case, the Galois group is the direct product of the Galois groups of the factors. It is cyclic of order 6. – i. m. soloveichik Dec 12 '12 at 22:32
You must have covered how to determine the Galois group of an arbitrary cubic. Does the word "discriminant" ring a bell? – Alex B. Dec 13 '12 at 13:57

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