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

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To determine the Galois group of the polynomial f(x)=x^3+x^2-2x-1 over the rationals, the discriminant of the cubic must be computed. If the discriminant is a square in the rational numbers, the Galois group is isomorphic to A3; if it is not a square, the group is isomorphic to S3. The automorphisms of the splitting field are defined by the permutations of the roots of the polynomial. Understanding these properties helps in identifying the structure of the Galois group. The discussion emphasizes the importance of the discriminant in classifying the Galois group.
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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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