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

  • Thread starter Thread starter mathmari
  • Start date Start date
  • Tags Tags
    Group
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
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)
 
Physics news on Phys.org
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 $S_3$.
 
Thread 'Derivation of equations of stress tensor transformation'
Hello ! I derived equations of stress tensor 2D transformation. Some details: I have plane ABCD in two cases (see top on the pic) and I know tensor components for case 1 only. Only plane ABCD rotate in two cases (top of the picture) but not coordinate system. Coordinate system rotates only on the bottom of picture. I want to obtain expression that connects tensor for case 1 and tensor for case 2. My attempt: Are these equations correct? Is there more easier expression for stress tensor...
Back
Top