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Concept
Artin reciprocity law
Summary
The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
Let be a Galois extension of global fields and stand for the idèle class group
of . One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol map
where denotes the abelianization of a group. The map is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the norm residue symbol
for different places of . More precisely, is given by the local maps on the -component of an idèle class. The maps are isomorphisms. This is the content of the local reciprocity law, a main theorem of local class field theory.
A cohomological proof of the global reciprocity law can be achieved by first establishing that
constitutes a class formation in the sense of Artin and Tate. Then one proves that
where denote the Tate cohomology groups. Working out the cohomology groups establishes that is an isomorphism.
Quadratic reciprocity and Eisenstein reciprocity
Artin's reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K which is based on the Hasse local–global principle and the use of the Frobenius elements. Together with the Takagi existence theorem, it is used to describe the abelian extensions of K in terms of the arithmetic of K and to understand the behavior of the nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of global class field theory. It can be used to prove that Artin L-functions are meromorphic, and also to prove the Chebotarev density theorem.
Official source
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