On Carmichael polynomials
type: Item
instance of: scholarly article
title: On Carmichael polynomials
zbMATH Open document ID: 0922.11105
author: Chih-Nung Hsu
published in: Journal of Number Theory
publication date: 1999-10-03T00:00:00Z
review text: Fermat's little theorem states that if \(l\) is a prime then \(l \mid a^l -a\) for all integers \(a\). A Carmichael number is a composite number such that \(n \mid a^n -a\) for all integers \(a\). It has recently been established that there are infinitely many such numbers. This paper considers the analogous situation with polynomials over the finite field \(F_q\), \(A = F_q [t]\). The role of the multiplicative group in the integer case is now played by the Carlitz module for the polynomial case. For any commutative \(A\)-algebra \(K\) let \(F^i\) be the \(q^i\) power Frobenius mapping. Let \(A\{ F \}\) be the \(A\)-submodule of \(A[x]\) generated by \(F^i ,i=0,1,2 \dots\). The Carlitz module \({\mathcal C}\) is then the unique \(F_q\)-linear ring homomorphism \(\phi: A \rightarrow A\{F\}\) given by: \[ \phi (1) = F^0 ,\quad\phi (t) = tF^0 +F^1 . \] A monic irreducible polynomial \(m \in A\) is called a Carmichael polynomial if it satisfies \(\phi (m-1)(\bar{a}) = (\bar{0}) \in {\mathcal C} (A/(m))\) for all \(a \in A\), where \(\bar{a}\) is the canonical image of \(a\) in \({\mathcal C} (A/(m))\). The paper shows that there exists infinitely many Carmichael polynomials for each \(q\). Using Carlitz modules of higher level \(n\), it is also shown that there exists Carmichael polynomials of level \(n\) for each \(q\) and \(n\).
Mathematics Subject Classification ID: 11T55
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 1236183
zbMATH Keywords: Carmichael polynomials
zbMATH Keywords: finite fields
zbMATH Keywords: Carlitz modules
reviewed by: Ian Blake
MaRDI profile type: Publication
full work available at URL: jnth.1998.2227
OpenAlex ID: W2019586721
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DOI: 10.1006/JNTH.1998.2227
description: scientific article; zbMATH DE number 1236183