Thread: modulo squares and cubes

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...

Originally Posted by lizzy

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...

Since , then for . Also, if is such a homomorphism, it must be that ,

where the product in the left side is modulo 7, whereas the sum in the right side is modulo 3.

But it must be .

From here you can now build your function taking into account, of course, that it must be since only are cubes in ...

Tonio