关于有限域上正规基的一个注记

A note on normal bases over finite fields

我们用初等方法证明了Chang等人在Journal of Algebra上发表的文章的主要结果:令p是一个素数,q是p的方幂,F_q是含q个元的有限域.若整数n≥2,则任一个n次首一非零迹的不可约多项式都是F_q上的正规多项式当且仅当n是p的方幂或n是一个不等于p的素数且q为n的一个原根.

Let q be a power of the prime p and q the q elements finite fields. In this note, by an elementa- ry way, the following result given by Chang et al in Journal of Algebra is proved. Let n ≥ 2 be an inte- ger, then every nth degree monic irreducible polynomial with non-zero trace is normal over q if and only if n is either a power of p or a prime number different from p as a primitive root.