Abstract: Given an odd prime number p, we characterize the partitions ℓ ̲ of p with p non negative parts ℓ 0 ≥ℓ 1 ≥...≥ℓ p-1 ≥0 for which there exist permutations σ,τ of the set {0,...,p-1} such that p divides ∑ i=0 p-1 iℓ σ(i) but does not divide ∑ i=0 p-1 iℓ τ(i) . This happens if and only if the maximal number of equal parts of ℓ ̲ is less than p-2. The question appeared when dealing with sums of p-th powers of resolvents, in order to solve a Galois module structure problem.