In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a norm group is a group of the form
where
is a finite
abelian extension In 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 a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
of nonarchimedean
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
s. One of the main theorems in
local class field theory In mathematics, 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 with respect to an absolute value or a discrete valuation with a finite re ...
states that the norm groups in
are precisely the open subgroups of
of finite index.
See also
*
Takagi existence theorem {{short description, Correspondence between finite abelian extensions and generalized ideal class groups
In class field theory, the Takagi existence theorem states that for any number field ''K'' there is a one-to-one inclusion reversing correspond ...
References
*J.S. Milne, ''Class field theory.'' Version 4.01.
Algebraic number theory
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