  Suppose $\mathbb{F}$ is a field of prime characteristic $p$ and $E$ is a
finite subgroup of the additive group $(\mathbb{F},+)$. Then $E$ is an
elementary abelian $p$-group. We consider two such subgroups, say $E$ and $E'$,
to be equivalent if there is an
$\alpha\in\mathbb{F}^*:=\mathbb{F}\setminus\{0\}$ such that $E=\alpha E'$. In
this paper we show that rational functions can be used to distinguish
equivalence classes of subgroups and, for subgroups of prime rank or rank less
than twelve, we give explicit finite sets of separating invariants.
