報告時間：2023年9月14日下午15:30-16:30

報告地點：kaiyun開云官方網站犀浦校區(qū)7教7510

報告人：洪紹方

摘要： Let $n\ge 1$ be an 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 whose degree is less than the maximal prime divisor of $c_0$. We also obtain all the possible degree of $f(x)$ which is reducible over $\Q$ when all the prime factor 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 Taylor polynomial 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, we show that the Galois group ${\rm Gal}_{\Q}(\E_{n,p})$ contains $A_{n}$ except for $(n,p)=(4,2)$, in which case, ${\rm Gal}_{\Q}(\E_{4,2})=S_3$. Finally, we show that the Galois group ${\rm Gal}_{\Q}(\E_{n,2})$ is $S_n$ if $n\equiv 3 \pmod 4$, or if $n$ is even and $v_q(n!)$ is odd for a prime divisor $q$ of $n-1$, or if $n\equiv 1\pmod 4$ and $n-2$ equals the product 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 a joint work with Dr. L.F. Ao.

報告人簡介：洪紹方，現任四川大學kaiyun開云官方網站教授，博士生導師。教育部新世紀優(yōu)秀人才，四川省學術和技術帶頭人。主要研究方向：數論，算術幾何和編碼理論。已經在國內外數學期刊上發(fā)表學術論文百余篇，已經培養(yǎng)畢業(yè)碩士、博士幾十名。