On the least primitive root in number fields  

作　　者：WANG TianZe1,& GONG Ke21School of Mathematics and Information Sciences,North China University of Water Resources and Electric Power,Zhengzhou 450011,China 2Department of Mathematics,Henan University,Kaifeng 475004,China 

出　　处：《Science China Mathematics》2010年第9期2489-2500,共12页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos.10671056,10801105)

摘　　要：Let K be an algebraic number field and OK its ring of integers.For any prime ideal p,the group(OK/p) of the reduced residue classes of integers is cyclic.We call any element of a generator of the group(OK/p) a primitive root modulo p.Stimulated both by Shoup's bound for the rational improvement and Wang and Bauer's generalization of the conditional result of Wang Yuan in 1959,we give in this paper a new bound for the least primitive root modulo a prime ideal p under the Grand Riemann Hypothesis for algebraic number field.Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.Let K be an algebraic number field and OK its ring of integers.For any prime ideal p,the group(OK/p) of the reduced residue classes of integers is cyclic.We call any element of a generator of the group(OK/p) a primitive root modulo p.Stimulated both by Shoup’s bound for the rational improvement and Wang and Bauer’s generalization of the conditional result of Wang Yuan in 1959,we give in this paper a new bound for the least primitive root modulo a prime ideal p under the Grand Riemann Hypothesis for algebraic number field.Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.

关 键 词：PRIMITIVE ROOT ALGEBRAIC number FIELDS HECKE ZETA function 

分 类 号：O156.2[理学—数学]