On the Galois group of generalized Laguerre polynomials
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 517-525.

En utilisant la théorie des polygones de Newton, on obtient un critère simple pour montrer que le groupe de Galois d’un polynôme soit “grand.” Si on fixe α- <0 , Filaseta et Lam ont montré que le Polynôme Generalisé de Laguerre L n (α) (x)= j=0 n n+α n-j(-x) j /j! est irréductible quand le degré n est assez grand. On utilise notre critère afin de montrer que, sous ces hypothèses, le groupe de Galois de L n (α) (x) est soit le groupe alterné, soit le groupe symétrique, de degré n, généralisant des résultats de Schur pour α=0,1,±1 2,-1-n.

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- <0 , Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L n (α) (x)= j=0 n n+α n-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 L n (α) (x) is either the alternating or symmetric group on n letters, generalizing results of Schur for α=0,1,±1 2,-1-n.

Publié le :
DOI : https://doi.org/10.5802/jtnb.505
Mots clés : Galois group, Generalized Laguerre Polynomial, Newton Polygon
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Farshid Hajir. On the Galois group of generalized Laguerre polynomials. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 517-525. doi : 10.5802/jtnb.505. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.505/

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