Non-vanishing of alternants

Let p be prime, K a field of characteristic 0. Let (x1, … , xn) ∈ Kn such that xi ≠ 0 for all i and xi/xj is not a root of unity for all i ≠ j. We prove that there exist integers 0 < e1 < e2 < ⋯ < en such that det(xjpei)≠0. The proof uses p-adic arguments. As corollaries, we derive the linear independence of certain Witt vectors and study the result of applying the Witt functor to a mod p representation of a finite group.