Primitive roots and powers among values of polynomials over finite fields.
Primitive roots and powers among values of polynomials over finite fields.
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有限域上多项式值的原根和幂。
DOI：
10.1515/crll.1984.350.137
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发表时间：
1984
期刊：
影响因子：
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通讯作者：
Stephen D. Cohen
中科院分区：
文献类型：
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作者：
Stephen D. Cohen
虽然Carlitz使用字符和实现渐近估计，马登和Velez得出了一个家庭的充分条件，通过任何素数权力q（即使是一个小的）可以测试。通过冗长的组合论证，他们证明了几乎所有的q都满足这些条件中的至少一个。事实上，在[9]中，他们用一些额外的劳动找到了包含所有例外奇q的有限集合A，当r = 2和n = 1或2时，它们适用于定理B。因此，如果Λ = 1，则A = {3，5，7}，而如果n = 2，则对于A '示出了具有最大成员631的26个元素的集合，实际上，素数103也应该被包括在内-它只是不满足[9]的推论1（iii）的条件。
While Carlitz used character sums to achieve asymptotic estimates, Madden and Velez derived a family of sufficient conditions by means of which any prime power q (even a small one) may be tested. By lengthy combinatorial arguments they showed that at least one of these conditions is met for almost all q. Indeed, in [9], with some extra labour they found finite sets A containing all exceptional odd q applicable in Theorem B when r = 2 and n = l or 2. Thus, if Λ = 1, then A = {3, 5, 7}, while, if « = 2, then a 26 element set with largest member 631 is shown for A', in fact, the prime 103 should also be included — it just fails to satisfy the condition of Corollary l (iii) of [9].