Shanta Laishram - On the Galois groups of generalized Laguerre Polynomials

hrj:1317 - Hardy-Ramanujan Journal, January 1, 2014, Volume 37 - https://doi.org/10.46298/hrj.2014.1317
On the Galois groups of generalized Laguerre Polynomials

Authors: Shanta Laishram

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.


Volume: Volume 37
Published on: January 1, 2014
Submitted 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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