The theory had its origins in the proof of quadratic reciprocity by gauss at the end of the 18th century. Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem. Local class field theory pan yan summer 2015 these are notes for a reading course with d. Class field theory faculty of science at bilkent university. There are undoubtably mistakes in these notes, and they are the authors alone. Class field theory is a branch of algebraic number theory that studies abelian extensions of number fields. Historically, local class field theory branched off from global, or classical. The origins of class field theory lie in the quadratic reciprocity law proved by gauss. Historically, local class field theory branched off from global, or classical, class field theory, which studies abelian extensions of global fieldsthat is. Classical class field theory and recent developments. In this thesis we formulate the main results of class field theory for local fields and for number fields. These ideas were developed over the next century, giving rise to a set of conjectures by hilbert. Recall that a global field is either a finite extension of characteristic 0 or a field of rational functions on a projective curve over a field of characteristic i.
A gentle course in local class field theory by pierre guillot. These are notes for a course in local class field theory taught at caltech winter term of 2008. In mathematics, local class field theory, introduced by helmut hasse, is the study of abelian extensions of local fields. The fractional kleingordon field or modified gravity inspired by quantum loop corrections. Class field theory describes the abelian extensions of a local or global field in terms of the arithmetic of the field itself.
This book offers a selfcontained exposition of local class field theory, serving as a second course on galois theory. We have put artins and tates theory of class formations at the beginning. Most parts of this can be found in ne2, iw and fevo1. An approach to time series analysis parzen, emanuel. A refinement of the local class field theory of serre and hazewinkel. Most of the terms and theorems stated here are needed later on, as we get to local class eld theory. Explicit class field theory for global function fields david zywina abstract. For the sake of formal simplicity we have used the notion of a. If you notice any mistakes or have any comments, please let me know. Class field theorythe bonn lectures newly edited by alexander schmidt author. But also its worth keeping in mind the following comment of miles reid from his undergraduate algebraic geometry p. This observation has a long history going back to fermat and euler. Advances in mathematics 18, 148181 1975 local class field theory is easy michiel hazewinkel department of math. Here the ultimate description is via quantum eld theory, but the classical approximation sometimes has widespread validity e.
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a nondiscrete topology. Local class field theory, strictly speaking, does not appear until the fourth part. The generalization took place as a longterm historical project, involving quadratic forms and their genus theory, work of ernst kummer and leopold kroneckerkurt hensel on ideals and completions, the theory of cyclotomic and kummer extensions. The abstract class field theory that we have developed in the last chapter is now going to be applied to the case of a local field, i. Local class field theory is a theory of abelian extensions of socalled local fields, typical examples of which are the padic number fields. It opens with a discussion of several fundamental topics in algebra, such as profinite groups, padic fields, semisimple algebras and their modules, and homological algebra with the example of group cohomology. Fields with local class field theory sciencedirect. While math 676 covered a variety of basic topics, math 776 will focus on a single topic.
Class field theory viewed as a langlands correspondence uvafnwi. Local class field theory, fall 2019 emory university. A nonarchimedean local eld of characteristic p0 is isomorphic to f qt for some power qof p. Computation of the norm residue symbol in certain local kummer fields 114 4. In the present paper, we su ggest yet another exp osition.
Collective behavior, from particles to fields cont. The tex file for this is here, and you will find hints in the comments. A global function eld is a nite extension of f pt for some prime p, or equivalently is the function eld of a geometrically integral curve over a nite eld f q called the constant eld, where qis a power of some prime p. The background below is meant to provide the early history and the motivation for constructing abelian extensions of number elds.
Course notes and supplementary material pdf format. This approach has the additional advantage of bringing out clearly the localglobal aspects of class. In mathematics, class field theory is the branch of algebraic number theory concerned with the abelian extensions of number fields, global fields of positive characteristic, and local fields. Over the rationals, for example, the theorem of kroneckerweber states that every abelian extension is contained in some cyclotomic field a number field generated by a suitable root of unity. Mar 12, 20 standard local class field theory is concerned with complete fields k whose residue field is finite. A nonarchimedean field k is a local field if it is locally compact with its metric topology. When general theory proves the existence of some construction, then doing it explicitly is a useful exercise that helps one to keep a grip on reality, but. Math 776, as the continuation of math 676, is a secondsemester graduate course in algebraic number theory. Ma432 classical field theory trinity college dublin.
Much of the material comes from keith conrads history of class eld theory. In this section we consider the isomorphism theorem. Local class eld theory says that there is a homomorphism. Field theoretic notions appeared, even though still implicitly, in the modern theory of solvability of polynomial equations, as introduced by abel and galois in the early nineteenth century. Introduction let k be a local field with finite residue field.
Computational class field theory universiteit leiden. Econometric institute, erasmus university, rotterdam, the netherlands 1. The author works out the consequences and the practical use of these results by giving detailed studies and illustrations of classical subjects. Let fbe a global function eld and let fab be its maximal abelian extension.
It begins with an analysis of the quadratic case of class field theory via hilbert symbols, in order to give a more handson introduction to the ideas of class field theory. This classic book, originally published in 1968, is based on notes of a yearlong seminar the authors ran at princeton university. If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web. Given such a field, an absolute value can be defined on it. Some pari programs have bringed out a property for the nongenus part of the class number of imaginary quadratic fields of fixed signature, with respect to vd. From theory to practice by george gras should have many examples. On the mean number of 2 torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields bhargava, manjul and varma, ila, duke. An overview of class field theory the statements of class eld theory become easier to understand once placed in their historical contexts. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory, and the authors accomplished this goal spectacularly.
The arithmetic of curves over two dimensional local fields draouil, belgacem, bulletin of the belgian mathematical society simon stevin, 2009. The first contains basic definitions and results on discrete valuation rings, dedekind domains which are their globalisation and the completion process. Introduction, saddle point approximation, and meanfield theory. More advanced topics in number theory are discussed in this course, such as galois cohomology, proofs of class field theory, modular forms and automorphic forms. Notes on class field theory notes from a onesemester course on class field theory uc berkeley, spring 2002. A global function eld is a nite extension of f pt for some prime p, or equivalently is the function eld of a geometrically integral curve over a nite eld f q called the constant eld, where qis. You should have a vague understanding of the use of complex multiplication to generate abelian extensions of imaginary quadratic fields first, in order. Hayes, we shall construct a continuous homomorphism galfabf. Our class field theory for curves over local fields corresponds to the case n 1 and dim x 1, and one of the most notable differences of the theory for n 2 1 from that for n 0 is the existence of nontrivial etale coverings of x in which all closed point split completely. In the first case, one calls the local field an archimedean local field. Class field theory, the study of abelian extensions of number fields.
Here is a more precise outline of the contents of these four parts. These notes contain an exposition of abelian class field theory using the algebraiccohomological approach of chevalley and artin and tate. A nonarchimedean local eld of characteristic 0 is isomorphic to a nite extension of q p. Cambridge core algebra a gentle course in local class field theory by pierre guillot. Class field theory for q first we discuss the situation where k q. The goal of this book is to present local class field theory from the cohomo logical point of view, following the method inaugurated by hochschild and developed. Local class field theory university of british columbia. In this case, we have the cyclotomic extension k q. On the other hand, class number formulas and the density theorems. Local number fields, brauer groups, galois cohomology. This theory includes the study of types of local fields, extensions of local fields using hensels lemma, galois extensions of local fields, ramification groups filtrations of galois groups of local fields, the behavior of the norm map on local fields, the local reciprocity homomorphism and existence theorem in local class field theory, local langlands correspondence. This is a oneyear course on class field theory one huge piece of intellectual work in the 20th century.