x^(2^ap^br^c)-1在有限域上的完全分解

Explicit Factorization of x^(2^αp^br^c)-1 over a Finite Field

F_q是阶为奇素数幂q的有限域.本文给出了x^(2^ap^br^c)-1在Fq中完全分解式,其中a,b,c均为正整数,p,r为q-1的两个不同的奇素数因子.结果表明x^(2^ap^br^c)-1在F_q上的所有不可约因子均为二项式或三项式.对一般情况,如果用v_p(m)表示正整数m的标准分解中素因子p的次数,假设m的每个素因子都整除q-1,那么:(1)当v_p(m)≤v_p(q-1)对任意素数p|q-1成立时,x^m-1在F_q上的不可约因子都是二项式;(2)当q≡3(mod 4)时,x^m-1在F_q上的不可约因子都是二项式或者三项式.

Let F_q be a finite field of odd order q.In this paper,the irreducible factorization of x^(2^a p^b r^c — 1 over F_q is given in a very explicit form,where a,b,c are positive integers and p,r are odd prime divisors of q — 1.It is shown that all the irreducible factors of x^(2a)p^br^c — 1 over F_q are either binomials or trinomials.In general,denote by v_p(m) the degree of prime p in the standard decomposition of the positive integer m.Suppose that every prime factor of m divides q — 1,one has:(1) if v_p(m) <v_p(q — 1) holds true for every prime number p | q — 1,then every irreducible factor of x^m — 1 in F_q is a binomial;(2) if q = 3(mod 4),then every irreducible factor of x^m — 1is either a binomial or a trinomial.