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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

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

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