The Quadratic Reciprocity Law of Duke-hopkins
نویسنده
چکیده
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 permutations and “quadratic symbols” in a more ambitious way. En route, we explore quadratic reciprocity as expressed in terms of the Kronecker symbol.
منابع مشابه
Quadratic Reciprocity in a Finite Group
The law of quadratic reciprocity is a gem from number theory. In this article we show that it has a natural interpretation that can be generalized to an arbitrary finite group. Our treatment relies almost exclusively on concepts and results known at least a hundred years ago. A key role in our story is played by group characters. Recall that a character χ of a finite Abelian group G is a homomo...
متن کاملGroup 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...
متن کاملQuadratic Reciprocity in Characteristic 2
Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] lets us decide whether or not a quadratic congruence f ≡ x2 mod π is solvable, where the modulus π is irreducible in F[T ] and f 6≡ 0 mod π. This is similar to the quadratic reciprocity law in Z. We want to develop an analogous reciprocity law when F has characteristic 2. At first it does not seem tha...
متن کاملPower Map Permutations and Symmetric Differences in Finite Groups
Let G be a finite group. For all a ∈ Z, such that (a, |G|) = 1, the function ρa : G → G sending g to g defines a permutation of the elements of G. Motivated by a recent generalization of Zolotarev’s proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation ρa. By introducing the group of conjugacy equivariant maps and the symmetric difference met...
متن کاملQuadratic Reciprocity in Odd Characteristic
The answer to questions like this can be found with the quadratic reciprocity law in F[T ]. It has a strong resemblance to the quadratic reciprocity law in Z. We restrict to F with odd characteristic because when F has characteristic 2 every element of F[T ]/(π) is a square, so our basic question is silly in characteritic 2. (There is a good analogue of quadratic reciprocity in characteristic 2...
متن کامل