I'm studying "The Arithmetic of Elliptic Curves" by Silverman and I'm having trouble with the torsion subgroup.
Corollary 6.4 at page 86 states that the subgroup $E[m]$ which is the $m$-torsion subgroup is isomorphic as a group to $Z/mZ \times Z/mZ$ if $m$ is coprime to the characteristic of the field.
While it's not clear, it seems logical that he is talking about an algebraically closed field.
What bugs me is that at page 175, proposition 5.4 he proves the same proposition for complex elliptic curves (using the corrispondence with lattices) without any acknowledgement of the existence of the previous proof which should be more general.
Is there a reason for this? Or is just a little oversight on the part of Silverman?