Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x）...Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.展开更多
基金supported by the National Fundamental Scientific Research Project of China(2004CB318000)the NSFC Grant 60276016
文摘Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.
