Cohomological Interpretation
Let Lv⁄Kv 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 CL stand for the idèle class group of L. The maps θv 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.
Read more about this topic: Artin Reciprocity Law