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Artin reciprocity law
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.〔Helmut Hasse, ''History of Class Field Theory'', in ''Algebraic Number Theory'', edited by Cassels and Frölich, Academic Press, 1967, pp. 266–279〕 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.
== Significance ==
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 for the proof of the Chebotarev density theorem.〔Jürgen Neukirch, ''Algebraische Zahlentheorie'', Springer, 1992, Chapter VII〕
Two years after the publication of his general reciprocity law in 1927, Artin rediscovered the transfer homomorphism of I. Schur and used the reciprocity law to translate the principalization problem for ideal classes of algebraic number fields into the group theoretic task of determining the kernels of transfers of finite non-abelian groups.〔.〕
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