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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.

enter image description here

It is not clear to me why you cannot transpose the red dots but you can transpose the green ones.

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    The discriminant tells the story; if the discriminant is a square, then the Galois group is $C_3$; if the discriminant is not a square, then the Galois group is $S_3$.2011-05-11
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    I think it largely comes down to what counts as "geometric". I could very well imagine there being some sort of high-powered arithmetic geometry explanation for the difference, but I don't think such an explanation (if there is one) would give any sort of visualizable clues as to whether a given cubic has $C_3$ or $S_3$2011-05-11
  • 0
    If the discriminant is square does that correspond to some lattice being square? I don't think so. I want to construct a geometric object from an arbitrary cubic (maybe some restrictions, but certainly not knowing their Galois group beforehand) that will show what is happening.2011-05-11

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