一类广义Schur型多项式的不可约性(英文)  

On the irreducibility of a class of generalized Schurtype polynomials

作　　者：尹轩睿 吴荣军 朱光艳 YIN Xuan-Rui;WU Rong-Jun;ZHU Guang-Yan(School of Mathematics,Sichuan University,Chengdu 610064,China;Chengdu No.7 High School,Chengdu 610000,China;School of Mathematics,Southwest Minzu University,Chengdu 610041,China;School of Teacher Education,Hubei Minzu University,Enshi 445000,China)

机构地区：[1]四川大学数学学院,成都610064 [2]成都市第七中学,成都610000 [3]西南民族大学数学学院,成都610041 [4]湖北民族大学教师教育学院,恩施445000

出　　处：《四川大学学报（自然科学版）》2023年第3期21-25,共5页Journal of Sichuan University(Natural Science Edition)

基　　金：西南民族大学科研启动金资助项目(RQD2021100);四川省自然科学基金(2022NSFSC1830)。

摘　　要：设n为一个正整数,a_(1),…,a_(n)均为整数.Schur通过素理想分解证明:当a_(n)=±1时,多项式1+a_(1)x+a_(2)x^(2)/2!+…+a_(n-1)(x^(n-1))/((n-1)!))+a_(n)x^(n)+n!是不可约多项式.随后,Coleman利用p-adic Newton多边形重新证明了Schur的结果.本文研究了一类特殊的广义Schur型多项式1+x+x^(2)/2+x^(3)/6+…+x^(p^(a))/((p^(a)(p^(a)-1))的不可约性.借助p-adic Newton多边形工具,本文基于局部整体原则证明:当p为素数,a为正整数时该多项式在有理数域上不可约.Let n be a positive integer and a_(1),…,a_(n)G Z.Schur proved that the polynomial 1+a_(1)x+a_(2)x^(2)/2!+…+a_(n-1)(x^(n-1))/((n-1)!))+a_(n)x^(n)+n!is irreducible over Q by using the factorization of prime ideal,where a_(n)=±1.Then Coleman reproved Schufs result by using the method of p-radic Newton polygon.In this paper,we study the irreducibility of the generalized Schurtype polynomial 1+x+x^(2)/2+x^(3)/6+…+x^(p^(a))/((p^(a)(p^(a)-1))By using the tool of pscdic Newton polygon and applying the local-global principle,we prove the irreducibility of this polynomial,where p is a prime number and a is a positive integer.

关 键 词：不可约 p-adic牛顿多边形 局部整体原则 

分 类 号：O156.2[理学—数学]