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2006•Unpublished venueRequires access

On Another Proof of Gauss' Quadratic Reciprocity Law

Pei Xiao-wen

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Abstract

Because all elements of the multiplicative group F~*_p are square ,Gauss by that has proved the quadratic reciprocity law(l/p)=(p/l)(-1)~e(l)e(p) of Legendre's symbol in the finite field.If S is a subset of F~*_p such that F~*_p is the disjoint union of S and-S,we can take S=1,2,…,p-12.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.

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What this paper is about

Because all elements of the multiplicative group F~*_p are square ,Gauss by that has proved the quadratic reciprocity law(l/p)=(p/l)(-1)~e(l)e(p) of Legendre's symbol in the finite field.If S is a subset of F~*_p such that F~*_p is the disjoint union of S and-S,we can take S=1,2,…,p-12.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.

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Available abstract

Because all elements of the multiplicative group F~*_p are square ,Gauss by that has proved the quadratic reciprocity law(l/p)=(p/l)(-1)~e(l)e(p) of Legendre's symbol in the finite field.If S is a subset of F~*_p such that F~*_p is the disjoint union of S and-S,we can take S=1,2,…,p-12.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.

Key concepts: Legendre symbol, Reciprocity law, Quadratic Gauss sum, Quadratic residue, Mathematics, Trigonometric substitution, Quadratic equation, Lemma (botany)

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