Partitions of nonzero elements of a finite field into pairs

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.