Dedekind

Discussions of the foundations of mathematics and their history are frequently restricted to logical issues in a narrow sense, or else to traditional problems of analytic philosophy. From Dedekind t

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remark

This thorough work presents the fundamental results of modular function theory as developed during the nineteenth and early-twentieth centuries. It features beautiful formulas and derives them using skillful and ingenious manipulations, especially classical methods often overlooked today. Starting with the work of Gauss, Abel, and Jacobi, the book then discusses the attempt by Dedekind to construct a theory of modular functions independent of elliptic functions. The latter part of the book explains how Hurwitz completed this task and includes one of Hurwitz's landmark papers, translated by the author, and delves into the work of Ramanujan, Mordell, and Hecke. For graduate students and experts in modular forms, this book demonstrates the relevance of these original sources and thereby provides the reader with new insights into contemporary work in this area.

The nineteenth century saw the paradoxes and obscurities of eighteenth-century calculus gradually replaced by the exact theorems and statements of rigorous analysis. It became clear that all analysis could be deduced from the properties of the real numbers. But what are the real numbers and why do they have the properties we claim they do? In this charming and influential book, Richard Dedekind (1831–1916), Professor at the Technische Hochschule in Braunschweig, showed how to resolve this problem starting from elementary ideas. His method of constructing the reals from the rationals (the Dedekind cut) remains central to this day and was generalised by Conway in his construction of the 'surreal numbers'. This reissue of Dedekind's 1888 classic is of the 'second, unaltered' 1893 edition.

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic.

The invention of ideals by Dedekind in the 1870s was well ahead of its time, and proved to be the genesis of what today we would call algebraic number theory. His memoir 'Sur la Theorie des Nombres Entiers Algebriques' first appeared in instalments in the 'Bulletin des sciences mathematiques' in 1877. This is a translation of that work by John Stillwell, who also adds a detailed introduction that gives the historical background as well as outlining the mathematical obstructions that Dedekind was striving to overcome. Dedekind's memoir gives a candid account of his development of an elegant theory as well as providing blow-by-blow comments as he wrestled with the many difficulties encountered en route. A must for all number theorists.

This book introduces algebraic number theory through the problem of generalizing 'unique prime factorization' from ordinary integers to more general domains. Solving polynomial equations in integers leads naturally to these domains, but unique prime factorization may be lost in the process. To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces, and modules. It was left to Emmy Noether to encapsulate the properties of rings that make unique prime factorization possible, in what we now call Dedekind rings. The book develops the theory of these concepts, following their history, motivating each conceptual step by pointing to its origins, and focusing on the goal of unique prime factorization with a minimum of distraction or prerequisites. This makes a self-contained easy-to-read book, short enough for a one-semester course.

本書在介紹數學基礎的歷史之後，系統講述現代數學主體的基礎 ----- ZFC集論，其中重點詳述四種數(自然數、實數、序數和基數)的理論。 實數的構造與傳統的Dedekind分割和Cantor基本序列等構造方法不同，採用了算術超濾分數構造法(這使“無窮大自然數”直接進入了數學)。 本書適合高等學校數學類專業本科生或研究生用作教材或教學參考書，也可供哲學和理工科其他專業的師生或研究人員參考。

This volume focuses on discussing the interplay between the analysis, as exemplified by the eta invariant and other spectral invariants, the number theory, as exemplified by the relevant Dedekind sums

The collected works, in German, of the groundbreaking mathematician Bernhard Riemann (1826-66) first appeared in 1876.

The great nineteenth-century mathematician Peter Gustav Lejeune Dirichlet (1805–59) studied in Paris, coming under the influence of scholars including Fourier and Legendre. He then taught at Berlin and Göttingen universities, where he was the successor to Gauss and mentor to Riemann and Dedekind. His achievements include the first satisfactory proof of the convergence of Fourier series under appropriate conditions, and the theorem on primes in arithmetic progression which was, at the same time, the foundation of analytic number theory and one of its greatest achievements. He also did important work on Laplace's equation, the theory of series and many other topics. This two-volume collection of his works, published 1889–97, was compiled by Leopold Kronecker (1823–91). Volume 1 contains works published by Dirichlet up to 1843 and a related 1846 essay. Volume 2, completed by Lazarus Fuchs (1833–1902), contains Dirichlet's later publications, unpublished papers and selected correspondence.

The great nineteenth-century mathematician Peter Gustav Lejeune Dirichlet (1805–59) studied in Paris, coming under the influence of scholars including Fourier and Legendre. He then taught at Berlin and Göttingen universities, where he was the successor to Gauss and mentor to Riemann and Dedekind. His achievements include the first satisfactory proof of the convergence of Fourier series under appropriate conditions, and the theorem on primes in arithmetic progression which was, at the same time, the foundation of analytic number theory and one of its greatest achievements. He also did important work on Laplace's equation, the theory of series and many other topics. This two-volume collection of his works, published 1889–97, was compiled by Leopold Kronecker (1823–91). Volume 1 contains works published by Dirichlet up to 1843, together with a related 1846 essay.

The great nineteenth-century mathematician Peter Gustav Lejeune Dirichlet (1805–59) studied in Paris, coming under the influence of scholars including Fourier and Legendre. He then taught at Berlin and Göttingen universities, where he was the successor to Gauss and mentor to Riemann and Dedekind. His achievements include the first satisfactory proof of the convergence of Fourier series under appropriate conditions, and the theorem on primes in arithmetic progression which was, at the same time, the foundation of analytic number theory and one of its greatest achievements. He also did important work on Laplace's equation, the theory of series and many other topics. This two-volume collection of his works, published 1889–97, was compiled by Leopold Kronecker (1823–91). Volume 2 was completed by Lazarus Fuchs (1833–1902) and contains Dirichlet's publications from 1844 onwards, together with some unpublished papers and selected correspondence with Gauss, Alexander von Humboldt and Kronecke

Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory, plus expander/Ramanujan graphs of interest in computer science. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are inclu

This text for a graduate-level course covers the general theory of factorization of ideals in Dedekind domains as well as the number field case. It illustrates the use of Kummer's theorem, proofs of t

This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read