28 citations · 121 across the 52 of their papers we have counts for
14 papers · 2 filters
Simerka - Quadratic Forms and Factorization
Franz Lemmermeyer
In this article we show that the Czech mathematician Vaclav Simerka discovered the factorization of (10^17-1)/9 using a method based on the class group of binary quadratic forms mo…
Harbingers of Artin's Reciprocity Law. I. The Continuing Story of Auxiliary Primes
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…
Counterexamples to the Hasse principle
Wayne Aitken, Franz Lemmermeyer
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 mi…
Arithmetic of Pell surfaces
Samuel Hambleton, Franz Lemmermeyer
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 discrimina…
Parametrizing Algebraic Curves
Franz Lemmermeyer
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 th…
Euclid's Algorithm in quartic CM-fields
Franz Lemmermeyer
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.
Relations in the 2-Class Group of Quadratic Number Fields
Franz Lemmermeyer
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…
Why is the Class Number of $\Q(\sqrt[3]{11})$ even?
Franz Lemmermeyer
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 t…
Jacobi and Kummer's Ideal Numbers
Franz Lemmermeyer
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 "Jacob…
Binary Quadratic Forms and Counterexamples to Hasse's Local-Global Principle
Franz Lemmermeyer
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 unsolvab…
Ideal class groups of cyclotomic number fields III
Franz Lemmermeyer
Using an idea going back to Scholz, we construct unramified abelian extensions of cyclotomic extensions of number fields.
Grün's Theorems and Class Groups
Franz Lemmermeyer
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.
Selmer Groups and Quadratic Reciprocity
Franz Lemmermeyer
In this article we study the 2-Selmer groups of number fields as well as some related groups, and present connections to the quadratic reciprocity law in .
Binomial Squares in Pure Cubic Number Fields
Franz Lemmermeyer
Let K = Q(ω) with ω^3 = m be a pure cubic number field. We show that the elementsα\in K^\times whose squares have the form a - ωform a group isomorphic to the group of rational poi…