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POLYNOMIAL EQUIVALENCE OF FINITE RINGS

Part of: Ring extensions and related topics Finite commutative rings Sequences and sets

Published online by Cambridge University Press:  01 April 2014

GEORG GRASEGGER ,
GÁBOR HORVÁTH  and
KEITH A. KEARNES
GEORG GRASEGGER
Affiliation:
Doctoral Program Computational Mathematics, Research Institute for Symbolic Computation, Johannes Kepler University Linz, Altenberger Strasse 69, 4040 Linz, Austria email Georg.Grasegger@risc.jku.at
GÁBOR HORVÁTH*
Affiliation:
Institute of Mathematics, University of Debrecen, Pf. 12, 4010 Debrecen, Hungary
KEITH A. KEARNES
Affiliation:
Department of Mathematics, University of Colorado, Boulder, CO 80309-0395, USA email Keith.Kearnes@Colorado.EDU
*
✉ghorvath@science.unideb.hu
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Abstract

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We prove that ${ \mathbb{Z} }_{{p}^{n} } $ and ${ \mathbb{Z} }_{p} [t] / ({t}^{n} )$ are polynomially equivalent if and only if $n\leq 2$ or ${p}^{n} = 8$. For the proof, employing Bernoulli numbers, we explicitly provide the polynomials which compute the carry-on part for the addition and multiplication in base $p$. As a corollary, we characterize finite rings of ${p}^{2} $ elements up to polynomial equivalence.

Keywords

polynomial equivalencefinite ringsBernoulli numberscarry for base p additioncarry for base p multiplication

MSC classification

Secondary: 13M10: Polynomials 13B25: Polynomials over commutative rings 11B68: Bernoulli and Euler numbers and polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 2 , April 2014 , pp. 244 - 257
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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