The quadratic reciprocity law and the elementary theta function

An Elementary Proof of the Law of Quadratic Reciprocity over Function Fields

Let P and Q be relatively prime monic irreducible polynomials in Fq [T ] (2 q). In this paper, we give an elementary proof for the following law of quadratic reciprocity in Fq [T ]: ( Q P )( P Q ) = (−1) |P |−1 2 |Q|−1 2 , where ( Q P ) is the Legendre symbol.

The Quadratic Reciprocity Law of Duke-hopkins

Circa 1870, G. Zolotarev observed that the Legendre symbol (ap ) can be interpreted as the sign of multiplication by a viewed as a permutation of the set Z/pZ. He used this observation to give a strikingly original proof of quadratic reciprocity [2]. We shall not discuss Zolotarev’s proof per se, but rather a 2005 paper of W. Duke and K. Hopkins which explores the connection between permutation...

Gauss Lemma and Law of Quadratic Reciprocity

The papers [20], [10], [9], [11], [4], [1], [2], [17], [8], [19], [7], [16], [13], [21], [22], [5], [18], [3], [15], [6], and [23] provide the terminology and notation for this paper. For simplicity, we adopt the following convention: i, i1, i2, i3, j, a, b, x denote integers, d, e, n denote natural numbers, f , f ′ denote finite sequences of elements of Z, g, g1, g2 denote finite sequences of ...

Group Theory and the Law of Quadratic Reciprocity

This paper explores the role of group theory in providing a proof for the Law of Quadratic Reciprocity, which states that for distinct odd primes pand q, q is a quadratic residue mod p if and only if p is a quadratic residue mod q, unless p and q are both congruent to 3 mod 4. The Law of Quadratic Reciprocity is an important result in number theory; it provides us with a simple method to determ...

The Historical Development of the Law of Quadratic Reciprocity