On the Babylonian Division of Trapezoids
We discuss the Babylonian division of trapezoids and the construction of such trapezoids with rational sides and transversals from Pythagorean triples.
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arXiv subjects
Franz Lemmermeyer
Publications and source records attributed to Franz Lemmermeyer.
Euclidean Rings
This is the text of my diploma thesis on Euclidean Rings from 1989, along with an English translation. Its main result is the generalization of Lenstra's idead of exceptional sequences to k-stage Euclidean rings.
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Kuroda's Class Number Formula
We give a non-analytic proof of the class number formula for biquadratic extensions of number fields.
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Arnold Scholz: Between Mathematics and Politics
This article provides information on the life and work of the number theorist Arnold Scholz. It is an English translation with modifications of an introduction to the correspondence of Hasse, Scholz and Taussky published in 2016.
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Vieta jumping and small norms in quadratic number fields
In this article we explain the connection between the famous problemfrom the IMO 1988 and elements of small norms in quadratic number fields with parametrized units.
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Gauss and p-adic numbers
In his notebooks, Gauss recorded various calculations with "infinite congruences". These infinite congruences are p-adic numbers; Gauss computes a square root of $5$ in the $11$-adic integers in order to find an $11$-adic approximation to a quadratic Gauss sum, computes a nontrivial square root of $1$ in $10$-adic integers, and computes the $10$-adic logarithms of small natural numbers.
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Octic Hilbert 2-class fields of real quadratic fields with discriminant 8p
In this article we explain how to construct cyclic octic unramfied extensions of the real quadratic number field $k = {\mathbb Q}(\sqrt{2p}\,)$, where $p \equiv 1 \bmod 8$ is a prime number such that $h_2(k) \equiv 0 \bmod 8$. The construction only requires solving the diophantine equation $eu^2 = t^2 + 2ps^2$ in integers.
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Extensions with Galois group Hol(C8) unramified over a complex quadratic number field
We study normal extensions with Galois group Hol($C_8$) that are unramified over a complex quadratic subfield. The Galois group is either the semi-dihedral group or the modular group of order $16$. We present an explicit construction of such fields.
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The Computation of Gal$(k^\infty/k)$ for some Complex Quadratic Number Fields $k$
We determine the Galois group of the 2-class field tower for two particular families of imaginary quadratic number fields $k$ with $2$-class field tower of length $2$.
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Dirichlet's Lemma in Number Fields
Dirichlet's Lemma states that every primitive quadratic Dirichlet character $\chi$ can be written in the form $\chi(n) = (\frac{\Delta}n)$ for a suitable quadratic discriminant $\Delta$. In this article we define a group, the separant class group, that measures the extent to which Dirichlet's Lemma fails in general number fields $F$. As an application we will show that over fields with trivial separant class groups, genus theory of quadratic extensions can be made as explicit as over the rationals.
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Hermite's identity and the quadratic reciprocity law
We give a short proof of the quadratic reciprocity law using Gauss's Lemma and Hermite's identity.
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Ein Brief von Eisensteins Eltern an Gauss
We give a transcription of a letter from Eisenstein's parents to Gauss, and an unpublished proof of the quadratic reciprocity law by Eisenstein using the tangent function.
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Gotthold Eisenstein and Philosopher John
Before the recent publication of the correspondence between Gauss and Encke, nothing was known about the role that John Taylor, a cotton merchant from Liverpool, had played in the life of Gotthold Eisenstein. In this article, we will bring together what we have discovered about John Taylor's life.
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Scholz's reciprocity law in number fields
In this article we prove a reciprocity law in number fields with odd class number that specializes to Scholz's reciprocity law over the rationals.
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On Leonardo da Vinci's proof of the Theorem of Pythagoras
We show that Leonardo da Vinci's well known proof of the Pythagorean theorem is due to Mayer and not to da Vinci.
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On 2-Class field towers of some imaginary quadratic number fields
We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2,2,2) whose Hilbert 2-class fields are finite.
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Rational Quartic Reciprocity
We provide a simple proof of the general rational quartic reciprocity law due to Williams, Hardy and Friesen.
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Rational Quartic Reciprocity II
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.
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