The following Cubics have 3 real roots but the first has Galois group $C_3$ and the second $S_3$
- $x^3 - 3x + 1$ (red)
- $x^3 - 4x + 2$ (green)
Is there any geometric way to distinguish between the two cases? Obviously graphing this onto the real line does not help.
It is not clear to me why you cannot transpose the red dots but you can transpose the green ones.