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2021 A Generalization of Bauer's Identical Congruence
Boaz COHEN
Tokyo J. Math. Advance Publication 1-28 (2021). DOI: 10.3836/tjm/1502179350

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.

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Boaz COHEN. "A Generalization of Bauer's Identical Congruence." Tokyo J. Math. Advance Publication 1 - 28, 2021. https://doi.org/10.3836/tjm/1502179350

Information

Published: 2021
First available in Project Euclid: 4 October 2021

Digital Object Identifier: 10.3836/tjm/1502179350

Subjects:
Primary: 11C08
Secondary: 11A07, 11R04, 11SXX

Rights: Copyright © 2021 Publication Committee for the Tokyo Journal of Mathematics

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28 PAGES

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