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On the Carmichael rings, Carmichael ideals and Carmichael polynomials

Volume 171 / 2023

Sunghan Bae, Su Hu, Min Sha Colloquium Mathematicum 171 (2023), 1-17 MSC: Primary 13A15; Secondary 11T06, 11R58, 11R60. DOI: 10.4064/cm8601-1-2022 Published online: 8 June 2022


Motivated by Carmichael numbers, we say that a finite ring $R$ is a Carmichael ring if $a^{|R|}=a$ for any $a \in R$. We then call an ideal $I$ of a ring $R$ a Carmichael ideal if $R/I$ is a Carmichael ring, and a Carmichael element of $R$ means it generates a Carmichael ideal. In this paper, we determine the structure of Carmichael rings and prove a generalization of Korselt’s criterion for Carmichael ideals in Dedekind domains. We extend several results from the number field case to the function field case. In particular, we study Carmichael elements of polynomial rings over finite fields (called Carmichael polynomials) by generalizing some classical results. For example, we show that there are infinitely many Carmichael polynomials but they have zero density.


  • Sunghan BaeDepartment of Mathematical Sciences
    Korea Advanced Institute
    of Science and Technology (KAIST)
    Daejeon 305-701, Republic of Korea
  • Su HuSchool of Mathematics
    South China University of Technology
    Guangzhou 510640, China
  • Min ShaSchool of Mathematical Sciences
    South China Normal University
    Guangzhou, 510631, China

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