arXiv subjects
Publications and source records attributed to Franz Lemmermeyer.
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 genus theory.
We construct a family of ideals representing ideal classes of order 2 in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.
In this note we present techniques to compute inhomogeneous minima of norm forms; as an application, we determine all norm-Euclidean complex bicyclic quartic number fields.
We present the technique of parametrization of plane algebraic curves from a number theorist's point of view and present Kapferer's simple and beautiful (but little known) proof that nonsingular curves of degree > 2 cannot be parametrized by rational functions.
We define a group stucture on the primitive integer points (A,B,C) of the algebraic variety Q_0(B,C)=A^n, where Q_0 is the principal binary quadratic form of fundamental discriminant Δand n is a fixed integer greater than 1. A surjective homomorphism is given from this group to the $n$-torsion subgroup of the narrow ideal class group of the quadratic number field Q(\sqrtΔ).
In this article we develop counterexamples to the Hasse principle using only techniques from undergraduate number theory and algebra. By keeping the technical prerequisites to a minimum, we hope to provide a path for nonspecialists to this interesting area of number theory. The counterexamples considered here extend the classical counterexample of Lind and Reichardt. As discussed in an appendix, this type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in the Tate--Shafarevich group.
In this article we give a modern interpretation of Kummer's ideal numbers and show how they developed from Jacobi's work on cyclotomy, in particular the methods for studying "Jacobi sums" which he presented in his lectures on number theory and cyclotomy in the winter semester 1836/37.
In this article we will describe a surprising observation that occurred in the construction of quadratic unramified extensions of a family of pure cubic number fields. Attempting to find an explanation will lead us on a magical mystery tour through the land of pure cubic number fields, Hilbert class fields, and elliptic curves.
We give a new proof of the theorem of Kronecker-Weber based on Kummer theory and Stickelberger's theorem.
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 $F$.
In this article we show how Grün's results in group theory can be used for studying the structure of class groups in normal extensions.
After a brief introduction to the classical theory of binary quadratic forms we use these results for proving (most of) the claims made by Pépin in a series of articles on unsolvable quartic diophantine equations, and for constructing families of counterexamples to the Hasse Principle for curves of genus 1 defined by equations of the form $ax^4 + by^4 = z^2$.
Using an idea going back to Scholz, we construct unramified abelian extensions of cyclotomic extensions of number fields.
We discuss the history of attempts to solve the Pell equation using certain auxiliary equations that correspond, in modern terminology, to a second 2-descent.
The aim of this article is to show that the arithmetic of Pell conics admits a description which is completely analogous to that of elliptic curves: there is a theory of 2-descent with associated Selmer and Tate-Shafarevich groups, and there should be an analog of the conjecture of Birch and Swinnerton-Dyer.
This article deals with the history of certain aspects of the Pell equation X$^2$ - D y$^2$ = 4. We briefly discuss explicit units, and then study the history of Legendre's equations ax$^2$ - b y$^2$ = 4 with ab=D.
In this articel we describe the first 2-descent on Pell conics, and compute the associated 2-part of the Tate-Shafarevich group.