Round trip through Mathematica's FullSimplify

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Conversion back from mathematica is a little dodgy at the moment. I would recommend applying the patch at #8495 if you want to call .sage() on different types of mathematica objects - otherwise just look at Mathematica's output directly (i.e. drop the .sage()) and retype the output.

However, in this case it doesn't seem to be anything to do with the conversion from Mathematica to Sage - Sage chooses this root even if you enter things directly:

sage: (-1)^(2/3)
1

I think this is because this is an even power.

sage: (-1)^(1/3)
(-1)^(1/3)

People complain about this all the time (including me, in the past) esp. because of plotting, though apparently this is consistent with Mma and Maple. But really it's the internal simplification in Ginac/Pynac that is doing this. Maxima, incidentally, does NOT do this.

(%i1) domain:complex;
(%o1)                               complex
(%i2) (-1)^(1/3);
                                        1/3
(%o2)                              (- 1)
(%i3) (-1)^(2/3);
                                        2/3
(%o3)                              (- 1)

I wonder what Mma and Maple do for this.