Deep congruences + the Brauer-Nesbitt theorem

We prove that mod-$p$ congruences between polynomials in $\mathbb{Z}_p[X]$ are equivalent to deeper $p$-power congruences between power-sum functions of their roots. This result generalizes to torsion-free $\mathbb{Z}_{(p)}$-algebras modulo divided-power ideals. Our approach is combinatorial: we introduce a $p$-equivalence relation on partitions, and use it to prove that certain linear combinations of power-sum functions are $p$-integral. We also include a second proof, short and algebraic, suggested by an anonymous referee. As a corollary we obtain a refinement of the Brauer-Nesbitt theorem for a single linear operator, motivated by the study of Hecke modules of mod-$p$ modular forms.