  Let $\mathcal{P}$ be an (unbounded) countable multiset of primes, let
$G=\bigoplus_{p\in P}\mathbb{F}_p$. We study the $k$'th universal
characteristic factors of an ergodic probability system $(X,\mathcal{B},\mu)$
with respect to some measure preserving action of $G$. We find conditions under
which every extension of these factors is generated by phase polynomials and we
give an example of an ergodic $G$-system that is not Abramov. In particular we
generalize the main results of Bergelson Tao and Ziegler who proved a similar
theorem in the special case $P=\{p,p,p,...\}$ for some fixed prime $p$. In a
subsequent paper we use this result to prove a general structure theorem for
ergodic $\bigoplus_{p\in P}\mathbb{F}_p$-systems.
