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

Partitions of nonzero elements of a finite field into pairs