POLYNOMIAL EQUIVALENCE OF FINITE RINGS

Abstract

AbstractWe 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.