A Generalization of Bauer's Identical Congruence

Abstract

In this paper we generalize Bauer's Identical Congruence appearing in Hardy and Wright's book [6], Theorems 126 and 127. Bauer's Identical Congruence asserts that the polynomial $\prod_t(x-t)$, where the product runs over a reduced residue system modulo a prime power $p^a$, is congruent (mod $p^a$) to the “simple” polynomial $(x^{p-1}-1)^{p^{a-1}}$ if $p>2$ and $(x^2-1)^{2^{a-2}}$ if $p=2$ and $a\geqslant2$. Our article generalizes these results to a broader context, in which we find a “simple” form of the polynomial $\prod_t(x-t)$, where the product runs over the solutions of the congruence $t^n\equiv 1\pmod{\mathrm{P}^a}$ in the framework of the ring of algebraic integers of a given number field $\mathbb{K}$, and where $\mathrm{P}$ is a prime ideal.