In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. [1]
Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. [2] The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem).
One of the major results is: given a number field F, and writing K for the maximal abelian unramified extension of F, the Galois group of K over F is canonically isomorphic to the ideal class group of F. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing CF for the idele class group of F, and taking L to be any finite abelian extension of F, this law gives a canonical isomorphism
where denotes the idelic norm map from L to F. This isomorphism is named the reciprocity map.
The existence theorem states that the reciprocity map can be used to give a bijection between the set of abelian extensions of F and the set of closed subgroups of finite index of
A standard method for developing global class field theory since the 1930s was to construct local class field theory, which describes abelian extensions of local fields, and then use it to construct global class field theory. This was first done by Emil Artin and Tate using the theory of group cohomology, and in particular by developing the notion of class formations. Later, Neukirch found a proof of the main statements of global class field theory without using cohomological ideas. His method was explicit and algorithmic.
Inside class field theory one can distinguish [3] special class field theory and general class field theory.
Explicit class field theory provides an explicit construction of maximal abelian extensions of a number field in various situations. This portion of the theory consists of Kronecker–Weber theorem, which can be used to construct the abelian extensions of , and the theory of complex multiplication to construct abelian extensions of CM-fields.
There are three main generalizations of class field theory: higher class field theory, the Langlands program (or 'Langlands correspondences'), and anabelian geometry.
In modern mathematical language, class field theory (CFT) can be formulated as follows. Consider the maximal abelian extension A of a local or global field K. It is of infinite degree over K; the Galois group G of A over K is an infinite profinite group, so a compact topological group, and it is abelian. The central aims of class field theory are: to describe G in terms of certain appropriate topological objects associated to K, to describe finite abelian extensions of K in terms of open subgroups of finite index in the topological object associated to K. In particular, one wishes to establish a one-to-one correspondence between finite abelian extensions of K and their norm groups in this topological object for K. This topological object is the multiplicative group in the case of local fields with finite residue field and the idele class group in the case of global fields. The finite abelian extension corresponding to an open subgroup of finite index is called the class field for that subgroup, which gave the name to the theory.
The fundamental result of general class field theory states that the group G is naturally isomorphic to the profinite completion of CK, the multiplicative group of a local field or the idele class group of the global field, with respect to the natural topology on CK related to the specific structure of the field K. Equivalently, for any finite Galois extension L of K, there is an isomorphism (the Artin reciprocity map)
of the abelianization of the Galois group of the extension with the quotient of the idele class group of K by the image of the norm of the idele class group of L.
For some small fields, such as the field of rational numbers or its quadratic imaginary extensions there is a more detailed very explicit but too specific theory which provides more information. For example, the abelianized absolute Galois group G of is (naturally isomorphic to) an infinite product of the group of units of the p-adic integers taken over all prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity. This is known as the Kronecker–Weber theorem, originally conjectured by Leopold Kronecker. In this case the reciprocity isomorphism of class field theory (or Artin reciprocity map) also admits an explicit description due to the Kronecker–Weber theorem. However, principal constructions of such more detailed theories for small algebraic number fields are not extendable to the general case of algebraic number fields, and different conceptual principles are in use in the general class field theory.
The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension (this is done inside local class field theory) and then prove that the product of all such local reciprocity maps when defined on the idele group of the global field is trivial on the image of the multiplicative group of the global field. The latter property is called the global reciprocity law and is a far reaching generalization of the Gauss quadratic reciprocity law.
One of the methods to construct the reciprocity homomorphism uses class formation which derives class field theory from axioms of class field theory. This derivation is purely topological group theoretical, while to establish the axioms one has to use the ring structure of the ground field. [4]
There are methods which use cohomology groups, in particular the Brauer group, and there are methods which do not use cohomology groups and are very explicit and fruitful for applications.
The origins of class field theory lie in the quadratic reciprocity law proved by Gauss. The generalization took place as a long-term historical project, involving quadratic forms and their 'genus theory', work of Ernst Kummer and Leopold Kronecker/Kurt Hensel on ideals and completions, the theory of cyclotomic and Kummer extensions.
The first two class field theories were very explicit cyclotomic and complex multiplication class field theories. They used additional structures: in the case of the field of rational numbers they use roots of unity, in the case of imaginary quadratic extensions of the field of rational numbers they use elliptic curves with complex multiplication and their points of finite order. Much later, the theory of Shimura provided another very explicit class field theory for a class of algebraic number fields. In positive characteristic , Kawada and Satake used Witt duality to get a very easy description of the -part of the reciprocity homomorphism.
However, these very explicit theories could not be extended to more general number fields. General class field theory used different concepts and constructions which work over every global field.
The famous problems of David Hilbert stimulated further development, which led to the reciprocity laws, and proofs by Teiji Takagi, Philipp Furtwängler, Emil Artin, Helmut Hasse and many others. The crucial Takagi existence theorem was known by 1920 and all the main results by about 1930. One of the last classical conjectures to be proved was the principalisation property. The first proofs of class field theory used substantial analytic methods. In the 1930s and subsequently saw the increasing use of infinite extensions and Wolfgang Krull's theory of their Galois groups. This combined with Pontryagin duality to give a clearer if more abstract formulation of the central result, the Artin reciprocity law. An important step was the introduction of ideles by Claude Chevalley in the 1930s to replace ideal classes, essentially clarifying and simplifying the description of abelian extensions of global fields. Most of the central results were proved by 1940.
Later the results were reformulated in terms of group cohomology, which became a standard way to learn class field theory for several generations of number theorists. One drawback of the cohomological method is its relative inexplicitness. As the result of local contributions by Bernard Dwork, John Tate, Michiel Hazewinkel and a local and global reinterpretation by Jürgen Neukirch and also in relation to the work on explicit reciprocity formulas by many mathematicians, a very explicit and cohomology-free presentation of class field theory was established in the 1990s. (See, for example, Class Field Theory by Neukirch.)
Class field theory is used to prove Artin-Verdier duality. [5] Very explicit class field theory is used in many subareas of algebraic number theory such as Iwasawa theory and Galois modules theory.
Most main achievements toward the Langlands correspondence for number fields, the BSD conjecture for number fields, and Iwasawa theory for number fields use very explicit but narrow class field theory methods or their generalizations. The open question is therefore to use generalizations of general class field theory in these three directions.
There are three main generalizations, each of great interest. They are: the Langlands program, anabelian geometry, and higher class field theory.
Often, the Langlands correspondence is viewed as a nonabelian class field theory. If and when it is fully established, it would contain a certain theory of nonabelian Galois extensions of global fields. However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case. It also does not include an analog of the existence theorem in class field theory: the concept of class fields is absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternatives to the Langlands correspondence point of view.
Another generalization of class field theory is anabelian geometry, which studies algorithms to restore the original object (e.g. a number field or a hyperbolic curve over it) from the knowledge of its full absolute Galois group or algebraic fundamental group. [6] [7]
Another natural generalization is higher class field theory, divided into higher local class field theory and higher global class field theory. It describes abelian extensions of higher local fields and higher global fields. The latter come as function fields of schemes of finite type over integers and their appropriate localizations and completions. It uses algebraic K-theory, and appropriate Milnor K-groups generalize the used in one-dimensional class field theory.
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and consequential conjectures about connections between number theory and geometry. Proposed by Robert Langlands, it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics."
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial splits into linear terms when reduced mod . That is, it determines for which prime numbers the relation
In mathematics, a global field is one of two types of fields that are characterized using valuations. There are two kinds of global fields:
In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.
In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbersQp (where p is any prime number), or the field of formal Laurent series Fq((T)) over a finite field Fq.
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers which is invariant under the action of a discrete subgroup of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated with a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.
Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity that generate a whole family of further number fields, analogously to the cyclotomic fields and their subfields. Leopold Kronecker described the complex multiplication issue as his liebster Jugendtraum, or "dearest dream of his youth", so the problem is also known as Kronecker's Jugendtraum.
The Artin reciprocity law, which was established by Emil Artin in a series of papers, is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far, only a small part of such a theory has been put on a firm basis.
In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of L-functions larger than Dirichlet L-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function.
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.
In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field K, to the general Galois extension L/K. While class field theory was essentially known by 1930, the corresponding non-abelian theory has never been formulated in a definitive and accepted sense.
In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map.
In algebraic number theory, the Shafarevich–Weil theorem relates the fundamental class of a Galois extension of local or global fields to an extension of Galois groups. It was introduced by Shafarevich for local fields and by Weil for global fields.
In mathematics, class field theory is the study of abelian extensions of local and global fields.
In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations.