In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions...

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...

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...

In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification...

geometric class field theory is an extension of class field theory to higher-dimensional geometrical objects: much the same way as class field theory describes...

In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and...

number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number...

module Galois cohomology Brauer group Class field theory Abelian extension Kronecker–Weber theorem Hilbert class field Takagi existence theorem Hasse norm...

which is finite, is called the class number of K {\displaystyle K} . The theory extends to Dedekind domains and their fields of fractions, for which the...

module theory of ideal class groups, initiated by Kenkichi Iwasawa (1959) (岩澤 健吉), as part of the theory of cyclotomic fields. In the early 1970s, Barry...

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...

conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the...

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...

the field K when the characteristic of K does divide n is called Artin–Schreier theory. Kummer theory is basic, for example, in class field theory and...

local data to global ones proves fruitful in class field theory, for example, where local class field theory is used to obtain global insights mentioned...

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory and the principle of relativity with ideas behind...

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously...

R×, and if R is a field, it is exactly the group of units. For a number field F, the group K2(F) is related to class field theory, the Hilbert symbol...

the field of formal Laurent series Fq((T)) over a finite field Fq, where q is a power of p. In particular, of importance in number theory, classes of local...

In sociology, field theory examines how individuals construct social fields, and how they are affected by such fields. Social fields are environments in...

mathematics, a ring class field is the abelian extension of an algebraic number field K associated by class field theory to the ring class group of some order...

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...

Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of...

algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local class field theory and diophantine...

theorem class field The class field theory concerns abelian extensions of number fields. class number 1.  The class number of a number field is the cardinality...

The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory. Algebraic...

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...

In algebraic number theory, a quadratic field is an algebraic number field of degree two over Q {\displaystyle \mathbf {Q} } , the rational numbers. Every...

Helmut Hasse and Richard Courant. He worked in mathematical logic and class field theory. He introduced recursive functions. Herbrand's theorem refers to either...

In number theory, the class number formula relates many important invariants of an algebraic number field to a special value of its Dedekind zeta function...