On the Galois group of Generalized Laguerre Polynomials

2004/06/15 by Farshid Hajir, Hajir, Farshid
Mathematics · #11R09 #11R32 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R09 #msc:11R32

paper · pdf · doi:10.48550/arxiv.math/0406308

6 pages

arxiv created 2004/06/15 · openalex publication_date 2004/06/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be ``large.'' For a fixed α∈ \Q - \Z_<0, Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L_n^(α)(x) = ∑_j=0ⁿ \binomn+αn-j(-x)^j/j! is irreducible for all large enough n. We use our criterion to show that, under these conditions, the Galois group of \La is either the alternating or symmetric group on n letters, generalizing results of Schur for α=0,1.

Citations

• Algebraic properties of a family of Generalized Laguerre Polynomials

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