Hardy-Ramanujan Journal |

For a positive integer n and a real number α, the generalized Laguerre polynomials are defined by L (α) n (x) = n j=0 (n + α)(n − 1 + α) · · · (j + 1 + α)(−x) j j!(n − j)!. These orthogonal polynomials are solutions to Laguerre's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for their interesting algebraic properties. In this short article, it is shown that the Galois groups of Laguerre polynomials L(α)(x) is Sn with α ∈ {±1,±1,±2,±1,±3} except when (α,n) ∈ {(1,2),(−2,11),(2,7)}. The proof is based on ideas of p−adic Newton polygons.

Source : oai:HAL:hal-01220303v1

Volume: Volume 37 - 2014

Published on: January 1, 2014

Imported on: October 30, 2015

Keywords: Laguerre Polynomials, Primes, Arithmetic Progressions, Newton Polygons,Irreducibility,[MATH] Mathematics [math],[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]

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