唯一分解整环R上不可约多项式的若干结构(英文).pdfVIP

Jou rnal of M athem atical R esearch Expo sit ion V o l. 19, N o. 2, 3672373, M ay 1999 Som e Structures of Irreduc ible Polynom ia ls over a Un ique Factor iza tion D oma in R Ξ W ang R u i (D ep t. of Comp. Sci. , Yunnan U niversity, Kunm ing 650091) Abstract: 　 In th is paper, w e give the concep tion of imp licit congruence and non imp licit congru2 ence in a un ique facto rizat ion dom ain R and estab lish som e structu res of irreducib le po lynom ials over R . A classical resu lt, E isenstein′s criterion, is generalized. Keywords: 　 un ique facto rizat ion dom ain, p rim e elem en t, non imp licit congruence, irreducib le po lynom ial. Cla ssif ica tion: AM S (1991) 11J ?CL C O 156 D ocum en t code: A 　 Article ID : (1999) 0220367207 1. In troduction 　　Early in the m iddle of the 19th cen tu ry, F. G. M. E isen stein, a Germ an m athem atician, gave th is fam ou s criterion [ 1, 2 ] : Theorem S upp ose tha t f (x ) = anx n + an21x n21 + ? + a1x + a0 is a p oly nom ia l w ith coef f icien ts in a un ique f actoriz a tion d om a in R. If there ex ists a p rim e elem en t p∈R such tha t 1°. p ??an; 2°. p ?a0, a1, ?, an21; 3°. p 2??a0, then f (x ) is irred ucible over R or its quotien t f ield. R ecen t ly, som eone show s [ 3 ] tha t E isen stein’s irreducib le condit ion is necessary and suf2 f icien t if the degree of f (x) is 2, on ly sufficien t if tha t of f (x) is grea ter than 2. T herefo re, it w ill be m ean ingfu l to im p rove and ex tend E isen stein’s theo rem. 2. M a in Results —763— Ξ Rece ived date: 1996208218; Rev ised date: 1999202228 B iography:W ang R u i(19602) ,m ale, bo rn in Tongx ian coun try, Beijing city. M. Sc. , cu rren t ly an associate p rofesso r at Yunnan U niversity. ? 1995-2005 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved. 　　L et R be a un ique facto riza t ion dom ain in th is paper. L emma 2. 1 S upp ose tha t f (x ) = anx n + an21x n21+ ?+ a1x + a0 is a p oly nom ia l w ith coef f icien ts in R ,