MHB What Determines the Galois Group of a Polynomial's Splitting Field?

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Hey! :o

We consider the polynomial $f(x)=x^3+x^2-2x-1 \in \mathbb{Q}[x]$ and let $E$ be its splitting field.

How can we find the group $Gal(E/\mathbb{Q})$ ?? (Wondering)
 
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Hi,

The automorphisms will be well defined with the image of the roots of $f$, and are also permutations over the roots, so you only have to check when a so defined automorphism is in the Galois group.
 
Fallen Angel said:
Hi,

The automorphisms will be well defined with the image of the roots of $f$, and are also permutations over the roots, so you only have to check when a so defined automorphism is in the Galois group.

Could you explain it further to me?? (Wondering)
 
Compute the discriminant of the cubic polynomial (it is irreducible). Then check if the discriminant is a square or not, in the square field. If it is a square the group is $A_3$, if it is a non-square then it is $S_3$.
 

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