On the Galois groups of generalized Laguerre Polynomials

Shanta Laishram (1)

1 Statistics and Mathematics Unit

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.

Mots clés: Primes; Arithmetic Progressions; Newton Polygons; Irreducibility; Laguerre Polynomials

Domaines: Mathématiques [math]; Théorie des nombres [math.NT]

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