# Determining the Galois Group of Splitting Field of Cubic polynomials

I wish to determine the Galois groups of $$L$$ over $$Q$$ when L is the splitting field of firstly, $$x^3 − 4x + 2$$ over $$Q$$ and secondly, $$x^3 − 3x + 1$$ over $$Q$$. For another example, I know $$x^3 + x^2 − 1$$. is irreducible by the rational root theorem. And the discriminant of the polynomial is $$−23$$, we have the group $$S_3$$ but I can't determine the Galois group for the above cubic polynomials in the same way, can I

• Can you work out the discriminants of your polynomials? – Lord Shark the Unknown Oct 16 '18 at 18:46
• opps yeah 148 and 81 respectively~ – Homaniac Oct 16 '18 at 19:03
• $148$ is not a square, but $81$ is. So the first has group $S_3$, and the second $A_3$. – Lord Shark the Unknown Oct 16 '18 at 19:44

In general: Let $$d$$ is discriminants polynomials.
if $$\sqrt d \in \Bbb Q$$ , then $$Gal(L, \Bbb Q)\cong A_3$$, while if $$\sqrt d \notin \Bbb Q$$ , then $$Gal(L, \Bbb Q)\cong S_3$$