In mathematics, **class field theory** is the study of abelian extensions of local and global fields.

- 1801 Carl Friedrich Gauss proves the law of quadratic reciprocity
- 1829 Niels Henrik Abel uses special values of the lemniscate function to construct abelian extensions of .
- 1837 Dirichlet's theorem on arithmetic progressions.
- 1853 Leopold Kronecker announces the Kronecker–Weber theorem
- 1880 Kronecker introduces his Jugendtraum about abelian extensions of imaginary quadratic fields
- 1886 Heinrich Martin Weber proves the Kronecker–Weber theorem (with a slight gap).
- 1896 David Hilbert gives the first complete proof of the Kronecker–Weber theorem.
- 1897 Weber introduces ray class groups and general ideal class groups.
- 1897 Hilbert publishes his Zahlbericht.
- 1897 Hilbert rewrites the law of quadratic reciprocity as a product formula for the Hilbert symbol.
- 1897 Kurt Hensel introduced
*p*-adic numbers. - 1898 Hilbert conjectures the existence and properties of the (narrow) Hilbert class field, proving them in the special case of class number 2.
- 1907 Philipp Furtwängler proves existence and basic properties of the Hilbert class field.
- 1908 Weber defines the class field of a general ideal class group.
- 1920 Teiji Takagi shows that the abelian extensions of a number field are exactly the class fields of ideal class groups.
- 1922 Takagi's paper on reciprocity laws
- 1923 Helmut Hasse introduced the Hasse principle (for the special case of quadratic forms).
- 1923 Emil Artin conjectures his reciprocity law.
- 1924 Artin introduces Artin L-functions.
- 1926 Nikolai Chebotaryov proves his density theorem.
- 1927 Artin proves his reciprocity law giving a canonical isomorphism between Galois groups and ideal class groups.
- 1930 Furtwängler and Artin prove the principal ideal theorem.
- 1930 Hasse introduces local class field theory.
- 1931 Hasse proves the Hasse norm theorem.
- 1931 Hasse classifies simple algebras over local fields.
- 1931 Jacques Herbrand introduces the Herbrand quotient.
- 1931 The Albert–Brauer–Hasse–Noether theorem proves the Hasse principle for simple algebras over global fields.
- 1933 Hasse classifies simple algebras over number fields.
- 1934 Max Deuring and Emmy Noether develop class field theory using algebras.
- 1936 Claude Chevalley introduces ideles.
- 1940 Chevalley uses ideles to give an algebraic proof of the second inequality for abelian extensions.
- 1948 Shianghao Wang proves the Grunwald–Wang theorem, correcting an error of Grunwald's.
- 1950 Tate's thesis uses analysis on adele rings to study zeta functions.
- 1951 André Weil introduces Weil groups.
- 1952 Artin and Tate introduce class formations in their notes on class field theory.
- 1952 Gerhard Hochschild and Tadashi Nakayama introduce group cohomology into class field theory.
- 1952 John Tate introduces Tate cohomology groups.
- 1964 Evgeny Golod and Igor Shafarevich prove that the class field tower can be infinite.
- 1965 Jonathan Lubin and Tate use Lubin–Tate formal group laws to construct ramified abelian extensions of local fields.

**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, **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.

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 **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 class field theory, the **Takagi existence theorem** states that for any number field *K* there is a one-to-one inclusion reversing correspondence between the finite abelian extensions of *K* and the **generalized ideal class groups** defined via a **modulus** of *K*.

In abstract algebra, an **adelic algebraic group** is a semitopological group defined by an algebraic group *G* over a number field *K*, and the adele ring *A* = *A*(*K*) of *K*. It consists of the points of *G* having values in *A*; the definition of the appropriate topology is straightforward only in case *G* is a linear algebraic group. In the case of *G* being an abelian variety, it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms.

**Kronecker's Jugendtraum** or **Hilbert's twelfth problem**, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogues of the roots of unity, as complex numbers that are particular values of the exponential function; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic fields and their subfields.

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.

This is a glossary of **arithmetic and diophantine geometry** in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.

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 mathematics, more specifically algebra, **abstract algebra** or **modern algebra** is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term *abstract algebra* was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Presently, the term "abstract algebra" is typically used for naming courses in mathematical education, and is rarely used in advanced mathematics.

The **Brumer–Stark conjecture** is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's theorem about the factorization of Gauss sums. It is named after Armand Brumer and Harold Stark.

In mathematics, the * Zahlbericht* was a report on algebraic number theory by Hilbert.

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 (1946) for local fields and by Weil (1951) for global fields.

In mathematics, a **ray class field** is an abelian extension of a global field associated with a **ray class group** of ideal classes or idele classes. Every finite abelian extension of a number field is contained in one of its ray class 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.

* Basic Number Theory* is an influential book by André Weil, an exposition of algebraic number theory and class field theory with particular emphasis on valuation-theoretic methods. Based in part on a course taught at Princeton University in 1961-2, it appeared as Volume 144 in Springer's

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*History of class field theory*(PDF) - Fesenko, Ivan, Class field theory, its three main generalisations, and applications, EMS Surveys in Mathematical Sciences 2021
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*Algebraic Number Theory*, Washington, D.C.: Thompson, pp. 266–279, MR 0218330 - Iyanaga, S. (1975) [1969], "History of class field theory",
*The theory of numbers*, North Holland, pp. 479–518 - Roquette, Peter (2001), "Class field theory in characteristic p, its origin and development",
*Class field theory—its centenary and prospect (Tokyo, 1998)*, Adv. Stud. Pure Math., vol. 30, Tokyo: Math. Soc. Japan, pp. 549–631

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