关于有限域中的元根

最近、王巨平使用数论中的Gauss和证明了:如果P~n≥Z~(60),则在有限域GF(P~n)中存在二个元根α和β,使得α+β=1。于是,Golomb有关元根的一个猜想基本上得到证明。本文用Jacobi和及王巨平提出的方法证明了若干更为一般的结论。此外,本文还基本上解决了Vegh提出的一个问题:是否对所有大于1的系数p,均能使得每一整数被表成P的二个元根之差。

ON PRIMITIVE ROOTS IN A FINITE FIELD

Recently, using elementary estimates for Gauss Sums, Ju- Ping Wang1] proved that if p>200 then there are two primitive roots x and y in GF(pn) such that x+y=6, where GF(pn) is a finlt?field,Using the method introduced in 1]and Jacobi sum, we prove following more gemeral theorems.Theorem 1. Suppose GF(p)is a finite field, where p is an odd prime mumher, {u, v, 0}then there are two primitive roots a and B in GF(p) such that ua+vB=6, where is Eulae's totient function, w(p-1) denotes the number of distinct prime factors of p- 1.Theorem 2 If p>200, then there are two primitive roots a and B Such that ua+vb=0.Theorem 3 If p>p(2w(9-1)2- (p- 1)((p-1)/(p-1) -1 , then there are twoprimitve roots a and |3 such that a~b=1.Vegh(2) asks whether, all primes p>61, every integer can be expressed as the difference of two primitive roots of p.From tneorem 2, we easily deduce following corollary.Corollary If p>280, then every integer can be expressed as the difference of two primitive roots of p.We can extend theorems 1-3 to GF(pn) (n>1) without difficult. For example, we have following theorem.Theorem 4 If pn>280 then there are two primitve roots a and b in GF(pn) such that ua+vb=0, where uv\0.