On the Galois groups of generalized Laguerre Polynomials

International audienceFor a positive integer $n$ and a real number $\alpha$, the generalized Laguerre polynomials are defined by\begin{align*}L^{(\alpha)}_n(x)=\sum^n_{j=0}\frac{(n+\alpha)(n-1+\alpha)\cdots (j+1+\alpha)(-x)^j}{j!(n-j)!}.\end{align*}These orthogonal polynomials are solutions to \emph{Laguerre's Differential Equation} which arises in thetreatment 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$L^{(\alpha)}_n(x)$ is $S_n$ with $\alpha\in \{\pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{1}{4}, \pm \frac{3}{4}\}$ except when $(\alpha, n)\in \{(\frac{1}{4}, 2), (-\frac{2}{3}, 11), (\frac{2}{3}, 7)\}$. The proof is based on ideas of $p-$adic Newton polygons