题  目：指数泰勒多项式的不可约性和伽罗瓦群

   主讲人：洪绍方 教授、博导

   时  间：10月28日（星期六）10:30-12:00  

   地  点：1-3

   主办单位：数学与计算机学院（大数据学院）

   主讲人简介：洪绍方，男，四川大学数学学院教授、博导，教育部新世纪优秀人才，四川省学术和技术带头人。任多家国际SCI期刊编委。主持国家自然科学基金和教育部博士点基金等纵向科研项目10多项，在30多种国内，国际著、知名数学期刊上发表学术论文100多篇，SCI收录论文90多篇。曾经多次访问美欧，以色列，日本，韩国等国以及香港和台湾等地区的著名高校和科研机构。已经培养毕业硕士60多名，毕业博士20余名，其中多人晋升正高职称。

   主讲内容：Let $n\ge 1$ bean integer and $f(x)=\frac{x^n}{n!}+\sum_{i=0}^{n-1}c_i\frac{x^i}{i!}$, where$c_0,c_1,...,c_{n-1}$ are arbitrary integers. In this talk,we show that if$f(x)$ is reducible over $\Q$, then there exists an irreducible factor whosedegree is less than the maximal prime divisor of $c_0$. We also obtain all thepossible degree of $f(x)$ which is reducible over $\Q$ when all the primefactor of $c_0$ is a subset of $\{2,3,5\}$. This extends a theorem of I. Schur.

   Let $p\in \{2,3,5\}$ and let$e_{n}(x):=\sum_{i=0}^n\frac{x^i}{i!}$ denote the truncated exponential Taylorpolynomial and $\E_{n,p}(x):=e_n(x)+(p-1)e_{n-1}(x)$. We prove that$\E_{n,p}(x)$ is irreducible if $(n,p)\not\in\{(2,2),(4,2)\}$. Furthermore, weshow that the Galois group ${\rm Gal}_{\Q}(\E_{n,p})$ contains $A_{n}$ exceptfor $(n,p)=(4,2)$, in which case,${\rmGal}_{\Q}(\E_{4,2})=S_3$. Finally, we show that the Galois group ${\rmGal}_{\Q}(\E_{n,2})$ is $S_n$ if $n\equiv 3 \pmod 4$, or if $n$ is even and$v_q(n!)$ is odd fora prime divisor $q$ of $n-1$, or if $n\equiv 1\pmod 4$ and $n-2$ equals theproduct of an odd prime number $l$ which is coprime to$\sum_{i=1}^{l-1}2^{l-1-i}i!$ and a positive integer coprime to $l$. This is ajoint work with Dr. L.F. Ao.