关于Hilbert符号的进一步讨论

On the Further Discussion of Hilbert Symbol

根据Hilbert定理及p进域的理论,推导出对于给定的Hilbert符号(a,b)有理数存在的充分必要条件,即对于Q*的一个有限元的簇(ai)i∈I和元素的值等于±1的簇(εi,v)i∈I,v∈V,存在有理数x∈Q*,对所有i∈I和v∈V,使(ai,x)v=εi,v成立的充分必要条件是满足以下3个条件:(1)对于几乎所有的εi,v都等于1;(2)对所有i∈I,有∏v∈Vεi,v=1;(3)对所有的v∈V,存在xv∈Qv*,使(ai,xv)v=εi,v成立(对所有的i∈I).

The reference[1]researched the calculation of the Hilbert symbol,and it gave the relationship between Hilbert symbol（a,b） and Legendre symbol（u p）.Based on Hilbert theorem and the theories of p-adic field,we reduce the necessary and sufficient about existence of rational numbers with giving Hilbert symbols,i e.let（ai）i∈I is a finite family of elements in Q＊ and let（εi,v）i∈I,v∈V is a family of numbers equal to ±1.In order that there exists x∈Q＊ such that（ai,x）v=εi,v for all i∈I and all v∈V,it is necessary and sufficient that the following conditions is satisfied.（1） Almost all the εi,v are equal to 1.（2） For all i∈I we have ∏ v∈V εi,v=1.（3） For all v∈V there exists xv∈Q＊v such that（ai,xv）v=εi,v for all i∈I in this paper.