  In this paper we prove that the nonzero elements of a finite field with odd
characteristic can be partitioned into pairs with prescribed difference (maybe,
with some alternatives) in each pair. The algebraic and topological approaches
to such problems are considered. We also give some generalizations of these
results to packing translates in a finite or infinite field, and give a short
proof of a particular case of the Eliahou--Kervaire--Plaigne theorem about
sum-sets.
