## Deep congruences + the Brauer-Nesbitt Theorem

**S. Anni, A. Ghitza, A. Medvedovsky**

26 pages, to appear in Integers.

We prove that mod-$p$ congruences between polynomials in $\mathbb Z_p[X]$ are equivalent to deeper mod-$p^{1+v_p(n)}$ congruences between the $n^{\rm th}$ power-sum functions of their roots. We give two proofs, one combinatorial and one algebraic. This result generalizes to torsion-free $\mathbb Z_{(p)}$-algebras modulo divided-power ideals. As a direct consequence, we obtain a refinement of the Brauer-Nesbitt theorem for finite free $\mathbb Z_p$-modules with an action of a single linear operator, with applications to the study of Hecke modules of mod-$p$ modular forms.