Talk:2.10.6/@comment-140.180.190.171-20120620030417

The first part of this proof describing the cosets of H is incorrect. As an example, consider z = 1+2i; the coset (1+2i)H is {1+2i, -1-2i, -2 + i, 2-i}, whereas the coset (1-2i)H is {1-2i, -1+2i, 2+i, -2-i}; these sets are not the same. One way to describe the cosets would be to identify the points in a given coset as the vertices of a square centred at the origin, or perhaps as the set of points obtained by repeatedly rotating a given point in the complex plane by pi/2 rad about the origin.

Also, a much easier way to do the second part is to let the function be f(z) = z^4.