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On the Galois groups of generalized Laguerre Polynomials

Abstract : 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.
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Shanta Laishram. On the Galois groups of generalized Laguerre Polynomials. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2014, Volume 37 - 2014, pp.8-12. ⟨10.46298/hrj.2014.1317⟩. ⟨hal-01220303⟩

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