nLab class field theory

Idea

Class field theory studies finite-dimensional abelian field extensions of number fields and of function fields, hence of global fields by relating them to the idele class group.

Class field theory clarifies the origin of various reciprocity laws in number theory. The basic (one dimensional) class field theory stems from the ideas of Kronecker and Weber, and results of Hilbert soon after them. Main results of the theory belong to the first half of the 20th century (Hilbert, Artin, Tate, Hasse…) and are quite different for the local field from the global field case. One of the basic objects the class group, is related to the Picard group in algebraic geometry. Generalizations for higher dimensional fields came later under now active higher class field theory, which is usually formulated in terms of algebraic K-theory and is closely related to deep questions of algebraic geometry (Paršin, Tate, Kato, Saito etc.).

Given an algebraic number field $k$ one defines a (congruence divisor) class field group $A/H$ in $k$; according to Weber

An algebraic extension $K/k$ is called a class field to $A/H$, if exactly those prime divisors in $k$ of first degree which belong to the principal class $H$ split completely in $K$

Some of the basic results of the class field theory are the Artin reciprocity theorem, existence theorem, uniqueness theorem, ordering theorem, Weber isomorphy theorem and the decomposition law of class field theory.

The Hilbert Class Field

For a number field $K$, the Hilbert class field $H$ is defined to be its maximal unramified abelian extension. In this special case, class field theory tells us that the Galois group $Gal(H/K)$ is isomorphic to the ideal class group of $K$.

In arithmetic topology, this is seen as an analogue of the Hurewicz theorem (see Morishita12).

Related concepts

• Langlands correspondence

• geometric class field theory

• Artin reciprocity law

References

• Albrecht Fröhlich, J. W. S. Cassels (eds.), Algebraic number theory, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, pdf, errata pdf by Kevin Buzzard)

• Albrecht Fröhlich, Martin J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, 1993

• wikipedia class field theory

• A. N. Paršin A. N. Local class field theory, Trudy Mat. Inst. Steklov 165 1984; Galois cohomology and Brauer group of local fields, Trudy Mat. Inst. Steklov 183, 1984.

The following survey of Connes-Marcolli work has a quick introduction to algebraic number theory including basic notions of CFT

• P. Almeida, Noncommutative geometry and arithmetics, Russian Journal of Mathematical Physics 16, No. 3, 2009, pp. 350–362, doi, see also nLab:arithmetic and noncommutative geometry

The following article sketches the geometric intuition behind the reciprocity laws as the relation between two approaches to the maximal abelian quotient of the fundamental group, mimicking the ideas of Galois theory

• Alexander Schmidt, Higher dimensional class field theory from a topological point of view, page

The following 2 articles make parallel between some notions of QFT and of number theory and in particular about the analogy between the Weil reciprocity law for function fields and the Takahashi-Ward identities of field theory:

• Leon Takhtajan, Quantum field theories on algebraic curves and A. Weil reciprocity law, arxiv/0812.0169; Quantum field theories on an algebraic curve, pdf, 2000

Some history is in

• James W. Cogdell, L-functions and non-abelian class field theory, from Artin to Langlands, 2012 (pdf)

The analogy between class field theory for the Hilbert class field and the Hurewicz theorem can be found in

• Masanori Morishita, Knots and Primes: An Introduction to Arithmetic Topology, 2012, Springer, (web)