Feasible alphabets for communicating the sum of sources over a network

We consider directed acyclic {\em sum-networks} with $m$ sources and $n$ terminals where the sources generate symbols from an arbitrary alphabet field $F$, and the terminals need to recover the sum of the sources over $F$. We show that for any co-finite set of primes, there is a sum-network which is solvable only over fields of characteristics belonging to that set. We further construct a sum-network where a scalar solution exists over all fields other than the binary field $F_2$. We also show that a sum-network is solvable over a field if and only if its reverse network is solvable over the same field.