eng
episciences.org
Hardy-Ramanujan Journal
2804-7370
2014-01-01
Volume 37 - 2014
10.46298/hrj.2014.1317
1317
journal article
On the Galois groups of generalized Laguerre Polynomials
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.
https://hrj.episciences.org/1317/pdf
Laguerre Polynomials
Primes
Arithmetic Progressions
Newton Polygons
Irreducibility
[MATH] Mathematics [math]
[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]