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51 elements of a field

Source: IMC 2003 day 1 problem 2

October 14, 2005
superior algebrasuperior algebra unsolved

Problem Statement

Let a1,a2,...,a51a_1, a_2,...,a_{51} be non-zero elements of a field of characteristic pp. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence b1,...,b51b_1, ... , b_{51}. If this new sequence is a permutation of the original one, find all possible values of pp.
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