class field theory (308 Ergebnisse)

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Taschenbuch. Zustand: Neu. Druck auf Anfrage Neuware - Printed after ordering - Research Paper (postgraduate) from the year 2011 in the subject Mathematics - Number Theory, grade: Postgraduate, University of Sheffield, language: English, abstract: This is the first in a two part series of papers establishing (with proof) the main theorems of global class field theory. We first recap some of the main ideas of algebraic number theory, using these to develop the Artin reciprocity law and the Takagi existence theorem both in terms of ideals and ideles. Finally, we use the Hilbert class field in order to study the well known problem of which prime numbers can be represented in the form x^2 + ny^2 for integers x,y and positive integer n.

Taschenbuch. Zustand: Neu. Neuware -Source: Wikipedia. Pages: 24. Chapters: Abelian extension, Albert¿Brauer¿Hasse¿Noether theorem, Artin L-function, Artin reciprocity law, Class formation, Complex multiplication, Conductor (class field theory), Galois cohomology, Genus field, Golod¿Shafarevich theorem, Grunwald¿Wang theorem, Hasse norm theorem, Hilbert class field, Hilbert symbol, Iwasawa theory, Kronecker¿Weber theorem, Lafforgue's theorem, Langlands dual, Langlands¿Deligne local constant, Local class field theory, Local Fields (book), Local Langlands conjectures, Non-abelian class field theory, Quasi-finite field, Takagi existence theorem, Tate cohomology group, Weil group. Excerpt: In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory. A formation is a topological group G together with a topological G-module A on which G acts continuously. A layer E/F of a formation is a pair of open subgroups E, F of G such that F is a finite index subgroup of E. It is called a normal layer if F is a normal subgroup of E, and a cyclic layer if in addition the quotient group is cyclic. If E is a subgroup of G, then A is defined to be the elements of A fixed by E. We write H(E/F)for the Tate cohomology group H(E/F, A) whenever E/F is a normal layer. (Some authors think of E and F as fixed fields rather than subgroup of G, so write F/E instead of E/F.) In applications, G is often the absolute Galois group of a field, and in particular is profinite, and the open subgroups therefore correspond to the finite extensions of the field contained in some fixed separable closure. A class formation is a formation such that for every normal layer E/F H(E/F) is trivial, andH(E/F) is cyclic of order |E/F|.In practice, these cyclic groups come provided with canonical generators uE/F ¿ H(E/F), called fundamental classes, that are compatible with each other in the sense that the restriction (of cohomology classes) of a fundamental class is another fundamental class. Often the fundamental classes are considered to be part of the structure of a class formation. A formation that satisfies just the condition H(E/F)=1 is sometimes called a field formation. For example, if G is any finite group acting on a field A, then this is a field formation by Hilbert's theorem 90. The most important examples of class formations (arranged roughly in order of difficulty) are as follows: It is easy to verify the class formation property for the finite field case and the archimedean local field case, butBooks on Demand GmbH, Überseering 33, 22297 Hamburg 24 pp. Englisch.

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Taschenbuch. Zustand: Neu. Neuware -Research Paper (postgraduate) from the year 2011 in the subject Mathematics - Number Theory, grade: Postgraduate, University of Sheffield, language: English, abstract: This document is a continuation of my Semester 1 project on class field theory. In the previous work, we made a rounded exposition of the fundamentals of class field theory but in order to preserve the document length the main proofs had to be skipped. We concentrate on filling in the gaps in this second installment. Due to the need to complete the arguments left open last semester and the need for applications this part of the project is a little longer than it should have been. It was not mentioned in the previous project but the class field theory we are studying here is global class field theory. There is such a thing as local class field theory in which we study the Abelian extensions of local fields (essentially fields that arise as completions of a number field with respect to places). Actually we touch on these ideas slightly in this project but never quite get to de_ning a local Artin map and looking at the local analogues of the main theorems of global class field theory. For those wanting to continue on to study local class field theory, consider Chapter 7 of [2] To start off this project we shall first restate the main de_nitions and theorems. This will be brief and those wanting to remind themselves of the details should consult my Semester 1 project. There will be very little motivation or technical results here since this was the purpose of the work done previously. We then set out to prove the main theorems of class field theory. With our present knowledge this would not be a simple task and we soon find that we first have to invent or discover new concepts such as the idele group and the corresponding idele class group. These are topological devices that take stock of all completions of a number eld at once. Such constructions will make the theory much easier to understand and formulate, whilst at the same time generalising the theory to all Abelian extensions. The cohomology of nite Abelian groups will be introduced and used alongside the idele theory to establish an important inequality. We use L-series in conjunction with the ideal theory to establish another important inequality. Combining the two inequalities will give a nice result that allows us to prove Artin reciprocity. In order to prove the existence theorem we resort to using Kummer n-extensions and the notion of a class eld. This middle chunk of the project will be quite technical but hopefully enjoyable and illuminating. [.]Books on Demand GmbH, Überseering 33, 22297 Hamburg 52 pp. Englisch.

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Soft cover. Zustand: New. This book, the second of three related volumes on number theory, is the English translation of the original Japanese book. Here, the idea of class field theory, a highlight in algebraic number theory, is first described with many concrete examples. A detailed account of proofs is thoroughly exposited in the final chapter. The authors also explain the local-global method in number theory, including the use of ideles and adeles. Basic properties of zeta and L-functions are established and used to prove the prime number theorem and the Dirichlet theorem on prime numbers in arithmetic progressions. With this book, the reader can enjoy the beauty of numbers and obtain fundamental knowledge of modern number theory. The translation of the first volume was published as Number Theory 1: Fermat's Dream, Translations of Mathematical Monographs (Iwanami Series in Modern Mathematics), vol. 186, American Mathematical Society, 2000.

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