Complete solving of explicit evaluation of Gauss sums in the index 2 case
Complete solving of explicit evaluation of Gauss sums in the index 2 case
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DOI：
10.1007/s11425-010-3155-z
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发表时间：
2009-11
期刊：
影响因子：
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通讯作者：
J. Yang;Lingli Xia
中科院分区：
文献类型：
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作者：
J. Yang;Lingli Xia
设p为素数，N为正整数，使得gcd（N，p）= 1，q= pf，其中ref为p模N的乘法阶。设χ是有限域上N阶本原乘法特征标.本文研究了Gauss和G（χ）在“指数2情形”（即[（χ/N χ）*：] = 2）下的显式求值问题.首先给出了指数为2的高斯和的分类。然后，在阶数N为一般偶数（即N= 2 r·N 0，其中N 0为正整数，N 0 <$3为奇数）的指数为2的情形下，得到了Gauss和G（χλ）（1 <$λ <$N− 1）的显式估计.这样，结合前人的研究，完全解决了指数2情形下高斯和的显式求值问题。
Letpbe a prime number,Nbe a positive integer such that gcd(N, p) = 1,q=pfwherefis the multiplicative order ofpmoduloN. Let χ be a primitive multiplicative character of orderNover finite field. This paper studies the problem of explicit evaluation of Gauss sumsG(χ) in the “index 2 case” (i.e. [(ℤ/Nℤ)*:] = 2). Firstly, the classification of the Gauss sums in the index 2 case is presented. Then, the explicit evaluation of Gauss sumsG(χλ) (1 ⩽ λ ⩽N− 1) in the index 2 case with orderNbeing general even integer (i.e.N= 2r·N0, wherer,N0are positive integers andN0⩾ 3 is odd) is obtained. Thus, combining with the researches before, the problem of explicit evaluation of Gauss sums in the index 2 case is completely solved.