Artin Reciprocity Law - Cohomological Interpretation

Cohomological Interpretation

Let L_v⁄K_v be a Galois extension of local fields with Galois group G. The local reciprocity law describes a canonical isomorphism

called the local Artin symbol.

Let L⁄K be a Galois extension of global fields and C_L stand for the idèle class group of L. The maps θ_v for different places v of K can be assembled into a single global symbol map by multiplying the local components of an idèle class. One of the statements of the Artin reciprocity law is that this results in the canonical isomorphism

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 θ is an isomorphism.