Leírás
Online Number Theory Seminar
Abstract: Let $\alpha$ and $n$ be integers with $n \geq 1.$ We shall discuss the problem of irreducibility of Generalized Laguerre Polynomial $L^{(\alpha)}_n(x)$ with degree $n$ and parameter $\alpha$ defined by
$$
L^{(\alpha)}_n(x)=\sum_{j=0}^n (-1)^{n-j}\frac{(n+\alpha)(n-1+\alpha)\cdots(j+1+\alpha)}{(n-j)!j!}x^j.
$$
When $\alpha=-n-1, L^{(-n-1)}_n(x)=\sum_{j=0}^n\frac{x^j}{j!}$ is the $n$th
Taylor polynomial of the exponential function. In 1930, I. Schur proved that $L^{(-n-1)}_n(x)$ is irreducible over the field $\mathbb{Q}$ of
rational numbers and its Galois group over $\mathbb{Q}$ is the alternating group $A_n$ or symmetric group $S_n$ of degree $n$ according as $n \equiv 0 \pmod{4}$ or not. We describe classes of the polynomials $L^{(\alpha)}_n(x)$ when these polynomials are irreducible over $\mathbb{Q}$. The irreducibility
is established using Newton polygons which will also be defined in the talk and their main properties together with some applications will be mentioned.
This talk is partly based on joint work with Shanta Laishram, Saranya Nair, Ritumoni Sarma and Tarlok Nath Shorey.
For access please contact the organizers (ntrg[at]science.unideb.hu).