On the distribution of powers in finite fields.
On the distribution of powers in finite fields.
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DOI：
10.1515/crll.1971.251.10
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发表时间：
1971
期刊：
影响因子：
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通讯作者：
J. Johnsen
中科院分区：
文献类型：
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作者：
J. Johnsen
在本文中，我们证明了存在任意长链的剩余给定的个人指数模一个足够大的素数。从而推广了A. Brauer [1]证明了连续A次方链的存在。特别地，我们得到了原根长度为t的链个数的一个渐近结果。我们还推出了著名的结果，存在二次剩余和非剩余的长链。在[4]和[5]中，E. Vegh用组合的方法处理了连续原根的问题。我们的结果是通过一些特征和的估计得到的。利用H. Hasse关于代数函数域的工作[3]和A.本文给出了关于曲线zeta函数零点的一个结果[6]。我们陈述我们的结果为大的有限域，并不限制自己的总理领域。最后，对任意r > l，证明了环Zjp的类似结果。我非常感谢莫里斯·戈德菲尔德博士，他给我展示了原根对数的初等证明，并提出了原根长链的问题。
In this paper we prove the existence of arbitrary long chains of residues of given individual indices modulo a sufficiently large prime. Thus we generalize a result of A. Brauer [1] who proved the existence of chains of consecutive A-th powers. In particular, we get an asymptotic result on the number of chains of length t of primitive roots. We also deduce the well known result that there exist long chains of quadratic residues and non-residues. In two papers, [4] and [5], E. Vegh has approached the problem of consecutive primitive roots by combinatorial means. Our results are obtained by means of estimates of some character sums. These estimates are proved by use of a well known method provided by H. Hasse's work on algebraic function fields [3] and A. We 's result on the zeros of the zeta-function of a curve [6]. We state our results for large finite fields and do not restrict ourselves to prime fields only. Finally, for any r > l, similar results are proved for the ring Zjp. l am very thankful to Dr. Morris Goldfeld who showed me an elementary proof for the number of pairs of primitive roots and suggested the problem of long chains of primitive roots.