On the Galois group of generalized Laguerre polynomials

Hajir, Farshid

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 $\alpha \in \mathbb{Q}-{\mathbb{Z}}_{<0}$, Filaseta et Lam ont montré que le Polynôme Generalisé de Laguerre $L_n^{\left(\alpha \right)}\left(x\right)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{n+\alpha }{n-j}\right){(-x)}^j/j!$ 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 $L_n^{\left(\alpha \right)}\left(x\right)$ est soit le groupe alterné, soit le groupe symétrique, de degré $n$, généralisant des résultats de Schur pour $\alpha =0,1,\pm \frac{1}{2},-1-n$.

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed $\alpha \in \mathbb{Q}-{\mathbb{Z}}_{<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^{\left(\alpha \right)}\left(x\right)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{n+\alpha }{n-j}\right){(-x)}^j/j!$ is irreducible for all large enough $n$. We use our criterion to show that, under these conditions, the Galois group of $L_n^{\left(\alpha \right)}\left(x\right)$ is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $\alpha =0,1,\pm \frac{1}{2},-1-n$.

Mots clés : Galois group, Generalized Laguerre Polynomial, Newton Polygon

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