Discussion Overview
The discussion revolves around determining the Galois group of the splitting field of the polynomial $f(x)=x^3+x^2-2x-1$ over the rational numbers. Participants explore methods related to automorphisms, discriminants, and the properties of the roots of the polynomial.
Discussion Character
- Technical explanation, Mathematical reasoning, Debate/contested
Main Points Raised
- One participant suggests that the automorphisms of the polynomial are defined by the images of its roots and that these automorphisms act as permutations of the roots, indicating a need to check which automorphisms belong to the Galois group.
- Another participant proposes computing the discriminant of the cubic polynomial and checking whether it is a square in the relevant field, stating that if the discriminant is a square, the Galois group is $A_3$, while if it is not a square, the group is $S_3$.
- A later reply requests further clarification on the initial explanation regarding automorphisms and their relation to the Galois group.
Areas of Agreement / Disagreement
Participants express different approaches to determining the Galois group, with no consensus reached on the method or the implications of the discriminant's properties.
Contextual Notes
The discussion does not resolve the assumptions regarding the irreducibility of the polynomial or the specific properties of the discriminant in relation to the Galois group.
