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. 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 L⁄K be a Galois extension of global fields and C_L stand for the idèle class group of L. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called global symbol map

The map ${\displaystyle \theta }$ is defined by assembling the maps called the local Artin symbol, the local reciprocity map or the norm residue symbol

for different places v of K. More precisely, ${\displaystyle \theta }$ is given by the local maps ${\displaystyle \theta _{v}}$ on the v-component of an idèle class. The maps ${\displaystyle \theta _{v}}$ are isomorphisms. This is the content of the local reciprocity law, a main theorem of local class field theory.

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