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 $S_3$.

 

[blue_raver22]

To find the group $Gal(E/\mathbb{Q})$, we can use Galois theory. First, we need to determine the degree of the extension $E/\mathbb{Q}$, which is equal to the degree of the minimal polynomial of any of the roots of $f(x)$. In this case, we can see that $f(x)$ has three distinct roots - $-1,1,$ and $2$. Therefore, the degree of the extension is $3$.

Next, we can use the fundamental theorem of Galois theory, which states that the order of the Galois group of an extension is equal to the degree of the extension. Since we know that the degree of our extension is $3$, we can conclude that the order of $Gal(E/\mathbb{Q})$ is also $3$.

To find the specific elements of the Galois group, we can use the fact that the Galois group is isomorphic to a subgroup of the symmetric group $S_3$ (since the degree is $3$). We can then use the properties of the roots of $f(x)$ to determine the permutations that correspond to the elements of the Galois group.

In summary, to find the group $Gal(E/\mathbb{Q})$, we need to determine the degree of the extension and then use Galois theory and the properties of the roots of $f(x)$ to determine the specific elements of the Galois group.