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

[mathmari]

Hey!

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)

 

[Fallen Angel]

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.

 

[mathmari]

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)

 

[ThePerfectHacker]

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 $S_3$.