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POLYNOMIAL EQUIVALENCE OF FINITE RINGS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 96 / Issue 2 / April 2014
- Published online by Cambridge University Press:
- 01 April 2014, pp. 244-257
- Print publication:
- April 2014
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- Article
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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.
