On the Galois group of generalized Laguerre polynomials

Farshid Hajir[1]

  • [1] Department of Mathematics & Statistics University of Massachusetts Amherst, MA 01003-9318 USA

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 2, page 517-525
  • ISSN: 1246-7405

Abstract

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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 n th 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 .

How to cite

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Hajir, Farshid. "On the Galois group of generalized Laguerre polynomials." Journal de Théorie des Nombres de Bordeaux 17.2 (2005): 517-525. <http://eudml.org/doc/249418>.

@article{Hajir2005,
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 $\alpha \in \mathbb\{Q\}- \mathbb\{Z\}_\{&lt;0\}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^\{(\alpha )\}(x) = \sum _\{j=0\}^n \binom\{n+\alpha \}\{n-j\}(-x)^j/j!$ is irreducible for all large enough $n$. We use our criterion to show that, under these conditions, the Galois group of $L_n^\{(\alpha )\}(x)$ is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $\alpha =0,1,\pm \frac\{1\}\{2\}, -1-n$.},
affiliation = {Department of Mathematics & Statistics University of Massachusetts Amherst, MA 01003-9318 USA},
author = {Hajir, Farshid},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Galois group; Generalized Laguerre Polynomial; Newton Polygon; Newton polygons},
language = {eng},
number = {2},
pages = {517-525},
publisher = {Université Bordeaux 1},
title = {On the Galois group of generalized Laguerre polynomials},
url = {http://eudml.org/doc/249418},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Hajir, Farshid
TI - On the Galois group of generalized Laguerre polynomials
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 2
SP - 517
EP - 525
AB - Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed $\alpha \in \mathbb{Q}- \mathbb{Z}_{&lt;0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre Polynomial $L_n^{(\alpha )}(x) = \sum _{j=0}^n \binom{n+\alpha }{n-j}(-x)^j/j!$ is irreducible for all large enough $n$. We use our criterion to show that, under these conditions, the Galois group of $L_n^{(\alpha )}(x)$ is either the alternating or symmetric group on $n$ letters, generalizing results of Schur for $\alpha =0,1,\pm \frac{1}{2}, -1-n$.
LA - eng
KW - Galois group; Generalized Laguerre Polynomial; Newton Polygon; Newton polygons
UR - http://eudml.org/doc/249418
ER -

References

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