If we look at the field F_7* modulo the squares, then we can say that this is isomorphic to Z/2 by sending the squares to 0, and the non-squares to 1 (at least we think this is the reason that it is isomorphic to Z/2).

Now we want to look at F_7* modulo cubes, we think this is isomorphic to Z/3, but we don't know why. We can send the cubes to 0, but how do we know which elements are mapped to 1 and 2?

We can't find anything about this on the web...