Abstract
We use the theory of Carlitz modules to define a sequence of linear polynomials \({\mathcal {L}}_{n,\phi }\), \(n\in {\mathbb {N}}\), that have properties similar to those satisfied by the classical Laguerre polynomials. We prove in particular that for any positive integer n, the Galois group over \({\mathbb {F}}_q(T)\) of \({\mathcal {L}}_{n,\phi }\) is the general linear group \(GL_n({\mathbb {F}}_q)\). This gives us explicit polynomials realizing \(GL_n({\mathbb {F}}_q)\) as a Galois group over \({\mathbb {F}}_q(T)\).
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The authors would like to thank the referee for all his valuable remarks, which allowed the improvement of our text and the clarification of our arguments.
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Oukhaba, H., El Kati, M. Laguerre Type Polynomials for Rational Function Fields and Applications. Mediterr. J. Math. 20, 72 (2023). https://doi.org/10.1007/s00009-023-02261-0
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DOI: https://doi.org/10.1007/s00009-023-02261-0