Summary: Let \(N_m=x^m+y^m\) be the \(m\)-th Newton polynomial in two variables, for \(m\geq 1\). Dvornicich and Zannier proved that in characteristic zero three Newton polynomials \(N_a, N_b, N_c\) are always sufficient to generate the symmetric field in \(x\) and \(y\), provided that \(a,b,c\) are distinct positive integers such that (a,b,c)=1. In the present paper we prove that in case of prime characteristic \(p\) the result still holds, if we assume additionally that \(a,b,c,a-b,a-c,b-c\) are prime with \(p\). We also provide a counterexample in the case where one of the hypotheses is missing.
The result follows from the study of the factorization of a generalized Vandermonde determinant in three variables, that under general hypotheses factors as the product of a trivial Vandermonde factor and an irreducible factor. On the other side, the counterexample is connected to certain cases where the Schur polynomials factor as a product of linear factors.