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On Carmichael polynomials. (English) Zbl 0922.11105
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 \(n\).

11T55 Arithmetic theory of polynomial rings over finite fields
11G09 Drinfel’d modules; higher-dimensional motives, etc.
Full Text: DOI
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