Abelian extension information

abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called...

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...

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements...

The motivating prototypical example of an abelian category is the category of abelian groups, Ab. Abelian categories are very stable categories; for...

{\displaystyle Q} to be abelian groups, then the set of isomorphism classes of extensions of Q {\displaystyle Q} by a given (abelian) group N {\displaystyle...

extension is Galois this automorphism group is called the Galois group of the extension. Extensions whose Galois group is abelian are called abelian extensions...

one-to-one inclusion reversing correspondence between the finite abelian extensions of K (in a fixed algebraic closure of K) and the generalized ideal...

– in what became known as the Kronecker Jugendtraum – that every abelian extension of K {\displaystyle K} could be obtained by the (roots of the) equation...

in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of...

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

of divergent series Abelian extension, in Galois theory, a field extension for which the associated Galois group is abelian Abelian von Neumann algebra...

regular representation. This is the case of the above, with Gal(K/Q) an abelian group, in which all the ρ can be replaced by Dirichlet characters (via...

that the first Ext group Ext1 classifies extensions of one module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer...

finite abelian extension of Q in C is contained in Q(ζn) for some n. Equivalently, the union of all the cyclotomic fields Q(ζn) is the maximal abelian extension...

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...

which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. A particularly...

In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces...

related to S-units of the field and a rational number. When K/k is an abelian extension and the order of vanishing of the L-function at s = 0 is one, Stark...

group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series...

(\exp(2\pi i/n))/\mathbb {Q} } is abelian, this is an abelian extension. Every subfield of a cyclotomic field is an abelian extension of the rationals. It follows...

In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which...

definition of the Artin map for a finite abelian extension L/K of global fields (such as a finite abelian extension of Q {\displaystyle \mathbb {Q} } ) has...