Number Fields
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Number Fields
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Author : Frans Keune
language : en
Publisher: Radboud University Press
Release Date : 2023-03-27
Number Fields written by Frans Keune and has been published by Radboud University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-03-27 with Mathematics categories.
Number Fields is a textbook for algebraic number theory. It grew out of lecture notes of master courses taught by the author at Radboud University, the Netherlands, over a period of more than four decades. It is self-contained in the sense that it uses only mathematics of a bachelor level, including some Galois theory. Part I of the book contains topics in basic algebraic number theory as they may be presented in a beginning master course on algebraic number theory. It includes the classification of abelian number fields by groups of Dirichlet characters. Class field theory is treated in Part II: the more advanced theory of abelian extensions of number fields in general. Full proofs of its main theorems are given using a ‘classical’ approach to class field theory, which is in a sense a natural continuation of the basic theory as presented in Part I. The classification is formulated in terms of generalized Dirichlet characters. This ‘ideal-theoretic’ version of class field theory dates from the first half of the twentieth century. In this book, it is described in modern mathematical language. Another approach, the ‘idèlic version’, uses topological algebra and group cohomology and originated halfway the last century. The last two chapters provide the connection to this more advanced idèlic version of class field theory. The book focuses on the abstract theory and contains many examples and exercises. For quadratic number fields algorithms are given for their class groups and, in the real case, for the fundamental unit. New concepts are introduced at the moment it makes a real difference to have them available.
The Theory Of Algebraic Number Fields
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Author : David Hilbert
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14
The Theory Of Algebraic Number Fields written by David Hilbert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.
Constance Reid, in Chapter VII of her book Hilbert, tells the story of the writing of the Zahlbericht, as his report entitled Die Theorie der algebra is chen Zahlkorper has always been known. At its annual meeting in 1893 the Deutsche Mathematiker-Vereinigung (the German Mathematical Society) invited Hilbert and Minkowski to prepare a report on the current state of affairs in the theory of numbers, to be completed in two years. The two mathematicians agreed that Minkowski should write about rational number theory and Hilbert about algebraic number theory. Although Hilbert had almost completed his share of the report by the beginning of 1896 Minkowski had made much less progress and it was agreed that he should withdraw from his part of the project. Shortly afterwards Hilbert finished writing his report on algebraic number fields and the manuscript, carefully copied by his wife, was sent to the printers. The proofs were read by Minkowski, aided in part by Hurwitz, slowly and carefully, with close attention to the mathematical exposition as well as to the type-setting; at Minkowski's insistence Hilbert included a note of thanks to his wife. As Constance Reid writes, "The report on algebraic number fields exceeded in every way the expectation of the members of the Mathemati cal Society. They had asked for a summary of the current state of affairs in the theory. They received a masterpiece, which simply and clearly fitted all the difficult developments of recent times into an elegantly integrated theory.
Algebraic Number Fields
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Author : Gerald J. Janusz
language : en
Publisher: American Mathematical Soc.
Release Date : 1996
Algebraic Number Fields written by Gerald J. Janusz and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Mathematics categories.
This text presents the basic information about finite dimensional extension fields of the rational numbers, algebraic number fields, and the rings of algebraic integers in them. The important theorems regarding the units of the ring of integers and the class group are proved and illustrated with many examples given in detail. The completion of an algebraic number field at a valuation is discussed in detail and then used to provide economical proofs of global results. The book contains many concrete examples illustrating the computation of class groups, class numbers, and Hilbert class fields. Exercises are provided to indicate applications of the general theory.
On The Class Number Of Abelian Number Fields
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Author : Helmut Hasse
language : en
Publisher: Springer
Release Date : 2019-04-23
On The Class Number Of Abelian Number Fields written by Helmut Hasse and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-04-23 with Mathematics categories.
With this translation, the classic monograph Über die Klassenzahl abelscher Zahlkörper by Helmut Hasse is now available in English for the first time. The book addresses three main topics: class number formulas for abelian number fields; expressions of the class number of real abelian number fields by the index of the subgroup generated by cyclotomic units; and the Hasse unit index of imaginary abelian number fields, the integrality of the relative class number formula, and the class number parity. Additionally, the book includes reprints of works by Ken-ichi Yoshino and Mikihito Hirabayashi, which extend the tables of Hasse unit indices and the relative class numbers to imaginary abelian number fields with conductor up to 100. The text provides systematic and practical methods for deriving class number formulas, determining the unit index and calculating the class number of abelian number fields. A wealth of illustrative examples, together with corrections and remarks on the original work, make this translation a valuable resource for today’s students of and researchers in number theory.
Class Groups Of Number Fields And Related Topics
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Author : Kalyan Chakraborty
language : en
Publisher: Springer Nature
Release Date : 2024-12-02
Class Groups Of Number Fields And Related Topics written by Kalyan Chakraborty and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-02 with Mathematics categories.
This book collects original research papers and survey articles presented at two conferences on the same theme: the International Conference on Class Groups of Number Fields and Related Topics, held at Kerala School of Mathematics, Kozhikode, Kerala, India, from 21–24 October 2021 and then from 21–24 November 2022. It presents the fundamental research problems that arise in the study of class groups of number fields and related areas. The book also covers some new techniques and tools to study these problems. Topics in this book include class groups of number fields, units, Ankeny–Artin–Chowla conjecture, Iwasawa theory, elliptic curves, Diophantine equations, partition functions, Diophantine tuples, congruent numbers, Carmichael ideals in a number field and their connection with class groups. This book will be a valuable resource for graduate students and researchers in mathematics interested in class groups of number fields and their connections to other branches of mathematics. It also attracts new researchers to the field and young researchers will benefit immensely from the diverse problems discussed in this book. All the contributing authors are leading academicians, scientists and profound researchers. This book is dedicated to Prof. Michel Waldschmidt, a renowned French number theorist, on his 75th birthday.
Fourier Analysis On Number Fields
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Author : Dinakar Ramakrishnan
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Fourier Analysis On Number Fields written by Dinakar Ramakrishnan and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.
This book grew out of notes from several courses that the first author has taught over the past nine years at the California Institute of Technology, and earlier at the Johns Hopkins University, Cornell University, the University of Chicago, and the University of Crete. Our general aim is to provide a modern approach to number theory through a blending of complementary algebraic and analytic perspectives, emphasizing harmonic analysis on topological groups. Our more particular goal is to cover Jolm Tate's visionary thesis, giving virtually all of the necessary analytic details and topological preliminaries-technical prereq uisites that are often foreign to the typical, more algebraically inclined number theorist. Most of the existing treatments of Tate's thesis, including Tate's own, range from terse to cryptic; our intent is to be more leisurely, more comprehen sive, and more comprehensible. To this end we have assembled material that has admittedly been treated elsewhere, but not in a single volume with so much detail and not with our particular focus. We address our text to students who have taken a year of graduate-level courses in algebra, analysis, and topology. While our choice of objects and methods is naturally guided by the specific mathematical goals of the text, our approach is by no means narrow. In fact, the subject matter at hand is germane not only to budding number theorists, but also to students of harmonic analysis or the representation theory of Lie groups.
Jacobi Forms Finite Quadratic Modules And Weil Representations Over Number Fields
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Author : Hatice Boylan
language : en
Publisher: Springer
Release Date : 2014-12-05
Jacobi Forms Finite Quadratic Modules And Weil Representations Over Number Fields written by Hatice Boylan and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-12-05 with Mathematics categories.
The new theory of Jacobi forms over totally real number fields introduced in this monograph is expected to give further insight into the arithmetic theory of Hilbert modular forms, its L-series, and into elliptic curves over number fields. This work is inspired by the classical theory of Jacobi forms over the rational numbers, which is an indispensable tool in the arithmetic theory of elliptic modular forms, elliptic curves, and in many other disciplines in mathematics and physics. Jacobi forms can be viewed as vector valued modular forms which take values in so-called Weil representations. Accordingly, the first two chapters develop the theory of finite quadratic modules and associated Weil representations over number fields. This part might also be interesting for those who are merely interested in the representation theory of Hilbert modular groups. One of the main applications is the complete classification of Jacobi forms of singular weight over an arbitrary totally real number field.
Cohomology Of Number Fields
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Author : Jürgen Neukirch
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-09-26
Cohomology Of Number Fields written by Jürgen Neukirch and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-09-26 with Mathematics categories.
This second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. In all it is a virtually complete treatment of a vast array of central topics in algebraic number theory. New material is introduced here on duality theorems for unramified and tamely ramified extensions as well as a careful analysis of 2-extensions of real number fields.
The Genus Fields Of Algebraic Number Fields
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Author : M. Ishida
language : en
Publisher: Springer
Release Date : 2006-12-08
The Genus Fields Of Algebraic Number Fields written by M. Ishida and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-12-08 with Mathematics categories.
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Foundations Of Analysis Over Surreal Number Fields
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Author : N.L. Alling
language : en
Publisher: Elsevier
Release Date : 1987-04-01
Foundations Of Analysis Over Surreal Number Fields written by N.L. Alling and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987-04-01 with Mathematics categories.
In this volume, a tower of surreal number fields is defined, each being a real-closed field having a canonical formal power series structure and many other higher order properties. Formal versions of such theorems as the Implicit Function Theorem hold over such fields. The Main Theorem states that every formal power series in a finite number of variables over a surreal field has a positive radius of hyper-convergence within which it may be evaluated. Analytic functions of several surreal and surcomplex variables can then be defined and studied. Some first results in the one variable case are derived. A primer on Conway's field of surreal numbers is also given.Throughout the manuscript, great efforts have been made to make the volume fairly self-contained. Much exposition is given. Many references are cited. While experts may want to turn quickly to new results, students should be able to find the explanation of many elementary points of interest. On the other hand, many new results are given, and much mathematics is brought to bear on the problems at hand.