关于高斯二次反比定律的另一种证明方法

On Another Proof of Gauss' Quadratic Reciprocity Law

高斯利用有限体中的元都是二次的这一性质,推证了Legendre符号的二次反比定律(l/p)=(p/l)(-1)ε(l)ε(p).如果乘群Fp*的子集S满足Fp*是S和-S的和,那么S可取为S=1,2,…,p-12的特殊形式.本文利用这一性质把Legendre符号的运算转化成S中元的三角运算,再通过三角引理的具体计算推得高斯二次反比定律,从而给出其另一种证明方法.

Because all elements of the multiplicative group Fp^＊ are square , Gauss by that has proved the quadratic reciprocity law （l/p）=（p/l）（-1）^ε（l）ε（p）. of Legendre＇ s symbol in the finite field. If S is a subset of F,; such that Fp^＊ is the disjoint union of S and - S, we can take S={ 1,2,..., （p- 1） /2} . In this paper, the author by using the proposition converted the operation of Legendre＇ s symbol into trigonometric operation . By the trigonometric lemma, we finally proved the Gauss＇ quadratic reciprocity law.