有限域上与仿射多项式有关的一些多项式的可约性

On the reducibility of some polynomials over finite fields with affine polynomials involved

在这篇文章中,研究了有限域上一些与仿射多项式有关的多项式的可约性.对于有限域Fp上不是xppt-x-1的仿射三项式,得到了这些三项式的一个明确的因式.完全确定了多项式g(xps-ax-b)在Fp[x]中的分解,这里g(x)是Fp[x]中一个不可约多项式.证明了Fp上次数相同的不可约多项式的全体可以构成一个正则图.同时给出了多项式g(xqs-x-b)在Fp[x]不可约因式的个数公式,这里g(x)是Fp上一个不可约多项式.

In this paper,the reducibility of some polynomials over finite fields with affine polynomials involved is investigated.For a given affine trinomial except for x^p^ pt -x-1 over Fp,an explicit factor of it is obtained. The decomposition of g（x^p s -ax-b）over Fp has been determined completely.It is also proved that the set of all irreducible polynomials over Fp with the same degree forms a regular graph.In addition,an explicit formula for the number of irreducible factors of g（x^q s -x-b）over Fp is presented,where g（x）is an irreducible polynomial over Fp.