Let B<sub>α</sub>（α）be an additive function on a ring of integers in the quadratic number field Q（（1/2）d）given by B<sub>α</sub>（α）=∑<sub>p丨α</sub><sup>*</sup...Let B<sub>α</sub>（α）be an additive function on a ring of integers in the quadratic number field Q（（1/2）d）given by B<sub>α</sub>（α）=∑<sub>p丨α</sub><sup>*</sup>N<sup>α</sup>（p）with a fixed α】0,where the asterisk means that the summation is over the non-associate prime divisors p of an integer α in Q（（1/2）d）,N（α）is the norm of α.In this paper we obtain the asymptotic formula of ∑<sub>N</sub>（α）≤<sub>x</sub> <sup>*</sup>B<sub>α</sub>（α）in the case where the class-number of Q（（1/2）d）is one.展开更多
In this paper, we discuss a method to compute the tame kernel of a number field. Confining ourselves to an imaginary quadratic field, we prove that _υ:K_2~S F/K_2~S F→k~* is bijective when N_υ>8δ_D^6.
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
Suppose F = Q(√-p1 pt) is an imaginary quadratic number field with distinct primes p1,..., pt,where pi≡ 1(mod 4)(i = 1,..., t- 1) and pt ≡ 3(mod 4). We express the possible values of the 8-rank r8 of the class grou...Suppose F = Q(√-p1 pt) is an imaginary quadratic number field with distinct primes p1,..., pt,where pi≡ 1(mod 4)(i = 1,..., t- 1) and pt ≡ 3(mod 4). We express the possible values of the 8-rank r8 of the class group of F in terms of a quadratic form Q over F2 which is defined by quartic symbols. In particular,we show that r8 is bounded by the isotropy index of Q.展开更多
基金Project supported by the National Natural Science Foundation of China.
文摘Let B<sub>α</sub>（α）be an additive function on a ring of integers in the quadratic number field Q（（1/2）d）given by B<sub>α</sub>（α）=∑<sub>p丨α</sub><sup>*</sup>N<sup>α</sup>（p）with a fixed α】0,where the asterisk means that the summation is over the non-associate prime divisors p of an integer α in Q（（1/2）d）,N（α）is the norm of α.In this paper we obtain the asymptotic formula of ∑<sub>N</sub>（α）≤<sub>x</sub> <sup>*</sup>B<sub>α</sub>（α）in the case where the class-number of Q（（1/2）d）is one.
文摘In this paper, we discuss a method to compute the tame kernel of a number field. Confining ourselves to an imaginary quadratic field, we prove that _υ:K_2~S F/K_2~S F→k~* is bijective when N_υ>8δ_D^6.
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.
基金supported by China Postdoctoral Science Foundation(Grant No.2013M541064)National Natural Science Foundation of China(Grant No.11371043)National Basic Research Program of China(Grant No.2013CB834202)
文摘Suppose F = Q(√-p1 pt) is an imaginary quadratic number field with distinct primes p1,..., pt,where pi≡ 1(mod 4)(i = 1,..., t- 1) and pt ≡ 3(mod 4). We express the possible values of the 8-rank r8 of the class group of F in terms of a quadratic form Q over F2 which is defined by quartic symbols. In particular,we show that r8 is bounded by the isotropy index of Q.
