Editing Global field (section)

===Artin reciprocity law===
{{main article|Artin reciprocity law}}
Artin's reciprocity law implies a description of the [[Commutator subgroup|abelianization]] of the absolute [[Galois group]] of a global field ''K'' that is based on the [[Hasse principle|Hasse local–global principle]]. It can be described in terms of cohomology as follows:

Let ''L''<sub>''v''</sub>/''K''<sub>''v''</sub> be a [[Galois extension]] of [[local field]]s with Galois group ''G''. The '''local reciprocity law''' describes a canonical isomorphism

: <math>\theta_v: K_v^{\times}/N_{L_v/K_v}(L_v^{\times}) \to G^{\text{ab}}, </math>

called the '''local Artin symbol''', the '''local reciprocity map''' or the '''norm residue symbol'''.{{sfn|Serre|1967|p=140}}{{sfn|Serre|1979|p=197}}

Let ''L''/''K'' be a [[Galois extension]] of global fields and ''C''<sub>''L''</sub> stand for the [[Adelic algebraic group|idèle class group]] 
of ''L''. The maps ''&theta;''<sub>''v''</sub> 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 a canonical isomorphism.{{sfn|Neukirch|1999|p=391}}{{sfn|Neukirch|1999|loc=Theorem 6.3|p=300}}