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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 $$n$$.

##### MSC:
 11T55 Arithmetic theory of polynomial rings over finite fields 11G09 Drinfel’d modules; higher-dimensional motives, etc.
##### Keywords:
Carmichael polynomials; finite fields; Carlitz modules
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##### References:
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