I'm reading the ECDSA paper and they say you can only use ECDSA with odd-power fields $p$ or with binary fields $2^m$. Why not other power prime fields?
Form a mathematical point of view, one can define ECDSA over arbitrary finite fields.
Form a security point of view, the most important thing is the size of the group order.
Arithmetic is easy in the case $GF(p)$.
Arithmetic gets more involved for $GF(p^m), m>1$, because you have to perform polynomial divisions.
Arithmetic in $GF(2^m)$ is easy again, because the polynomial division in $GF(2^m)$ can be done by simple feedback shift registers. This makes it especially well suited for implementation in hardware.