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

transformation, a gauge transformation Non-abelian class field theory, in class field theory Nonabelian cohomology, a cohomology Abelian (disambiguation) All pages...

gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory. Many powerful theories in physics are described by...

of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups...

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...

particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic...

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...

étale cohomology theory (roughly speaking, the theory as it applies to zero-dimensional schemes). Secondly, non-abelian class field theory was launched as...

"visionary". A larger field sometimes called arithmetic of abelian varieties now includes Diophantine geometry along with class field theory, complex multiplication...

mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to...

an abelian group. Every finite extension of a finite field is a cyclic extension. Class field theory provides detailed information about the abelian extensions...

absolute Galois groups of number fields and mixed-characteristic local fields. Section conjecture Class field theory Fiber functor Neukirch–Uchida theorem...

numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading...

elements. Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set. In order theory (and especially its...

category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major...

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...

compact Lie group. A Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of...

{-5}})} . Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning...

interaction, is a non-Abelian gauge theory with an SU(3) gauge symmetry. It contains three Dirac fields ψi, i = 1,2,3 representing quark fields as well as eight...

is abelian are called abelian extensions. For a given field extension L / K {\displaystyle L/K} , one is often interested in the intermediate fields F...

mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with...

and number theory with his doctoral work leading to the Mordell–Weil theorem which demonstrates that the set of rational points of an abelian variety is...

algebraic numerical coefficients 12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality 13. Impossibility of the solution...

developed by Jakob Nielsen Non-abelian class field theory Non-classical analysis Non-Euclidean geometry Non-standard analysis Non-standard calculus Nonarchimedean...

since it makes explicit class field theory in the way the roots of unity do for abelian extensions of the rational number field, via Shimura's reciprocity...

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...

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...

sequence of abelian groups called homology groups. This operation, in turn, allows one to associate various named homologies or homology theories to various...

center of any non-trivial finite p-group is non-trivial. If the quotient group G/Z(G) is cyclic, G is abelian (and hence G = Z(G), so G/Z(G) is trivial)...

Abelian extension Transcendence degree Field norm Field trace Conjugate element (field theory) Tensor product of fields Types Algebraic number field Global...