Related papers: Scholz's reciprocity law in number fields

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of…

We prove a reciprocity law for one-dimensional compatible systems of mod p representations of absolute Galois groups of number fields. We prove that these arise from Hecke characters, and in particular recover by purely algebraic means the…

Let $d$ be an odd square-free integer, $m\geq 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic…

The reciprocal Pascal matrix is the Hadamard inverse of the symmetric Pascal matrix. We show that the ordinary matrix inverse of the reciprocal Pascal matrix has integer elements. The proof uses two factorizations of the matrix of super…

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the…

In this paper, we study properties of nodal orders defined over arbitrary base fields. In particular we give a classification of complete real nodal orders.

The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces, and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve,…

In this article we present the history of auxiliary primes used in proofs of reciprocity laws from the quadratic to Artin's reciprocity law. We also show that the gap in Legendre's proof can be closed with a simple application of Gauss's…

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 prove non-commutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws claim the splittings of some central extensions of globally constructed groups over some subgroups constructed by points…

The paper proves the Strong Law of Large Numbers for integral functionals of random fields with unboundedly increasing covariances. The case of functional data and increasing domain asymptotics is studied. Conditions to guarantee that the…

This article is the first article of a serie of articles on the generalization of Kato's Euler system. The main subject of this article is to construct a family of Kato's Euler systems and a family of Kato's explicit reciprocity laws over…

We prove an interpolation result for homogeneous polynomials over the integers, or more generally for PIDs with finite residue fields. Previous proofs of this result use the well-known but nontrivial fact that class groups of rings of…

In this paper, using what we call a micro reciprocity law, we complete Weil's program for non-abelian class field theory of Riemann surfaces.

We study the bias of random bounded-degree polynomials over odd prime fields and show that, with probability exponentially close to 1, such polynomials have exponentially small bias. This also yields an exponential tail bound on the weight…

In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.

We prove new explicit conditional bounds for the residue at $s=1$ of the Dedekind zeta-function associated to a number field. Our bounds are concrete and all constants are presented with explicit numerical values.

Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.

Natural numbers satisfying an unusual property are mentioned by the author in [5], in which their infinitude is also proved. In this paper, we start with an arbitrary natural number which is not a multiple of 10 and non-palindromic, form…

We study odd numbers through a straightforward indexing. We focus in particular on odd prime and composite numbers and their distribution. With a counting argument, we calculate the limit of two sums and compare their convergence rate.