Related papers: Quadratic reciprocity in a finite group

We present a new proof of the celebrated quadratic reciprocity law. Our proof is based on group theory.

Using the quadratic reciprocity law as the motivating example, we convey an understanding of classical reciprocity laws.

Much has been written on reciprocity laws in number theory and their connections with group representations. In this paper we explore more on these connections. We prove a "reciprocity Law" for certain specific representations of semidirect…

In this article we study the 2-Selmer groups of number fields $F$ as well as some related groups, and present connections to the quadratic reciprocity law in $F$.

We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the…

Cubic and biquadratic reciprocity have long since been referred to as "the forgotten reciprocity laws", largely since they provide special conditions that are widely considered to be unnecessary in the study of number theory. In this…

Rousseau's simple proof of the quadratic reciprocity law, followed by the proof of its equivalence with Hilbert's product formula. The Hilbert symbol is explained in terms of the reciprocity isomorphism, and the places of Q are determined.

In 1991, Rousseau gave a new proof of Gauss's quadratic reciprocity by comparing two distinct coset representations of the group $(\mathbb{Z}_{p}^{*} \times \mathbb{Z}_{q}^{*}) / U$ using the Chinese Remainder Theorem, without Gauss's…

We give a short proof of the quadratic reciprocity law using Gauss's Lemma and Hermite's identity.

The shortest known proof of the law of quadratic reciprocity (without supplements) is presented.

We continue investigating rational quartic reciprocity laws and, at the suggestion of the editor of AA, provide details of a proof of a remark in the first article with this title.

We provide an algorithm which, for a given quadratic equation in the Grigorchuk group determines if it has a solution. As a corollary to our approach, we prove that the group has a finite commutator width.

In this note we will present a supplement to Scholz's reciprocity law and discuss applications to the structure of 2-class groups of quadratic number fields.

We obtain a lifting property for finite quotients of algebraic groups, and applications to the structure of these groups.

We construct a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and prove a simple…

In this paper we give an algorithm to determine all finite matrix groups over a number field. Our algorithm is based on the representation theory of finite groups.

The goal of this survey is to introduce all the necessary concepts and theorems to provide a rigorous and self-contained proof of the Law of Quadratic Reciprocity and see how this is a useful tool to obtain results such as the problem of…

We announce a very general statement involving the rational quartic residue symbol $(m/p)_4$ and, more generally, Legendre symbols of the type ${a+b\sqrt{m}/p$. We show how our main theorem can be used to produce many older results such as…

We revisit Eisenstein's geometric proof of quadratic reciprocity and make explicit the involutive symmetry underlying Eisenstein's lattice-point argument. Building on Gauss's lemma, we interpret the Legendre symbols as counts of lattice…

The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum…