The quadratic reciprocity law ties together pairs of prime numbers p, q in the beautiful formula (p|q)(q|p)=(-1)^(p-1)(q-1)/4 where (p|q) is the Legendre symbol. Algebraically, quadratic reciprocity law provides a way to determine if a congruence x^2 = a (mod p) is solvable. On the other hand, Artin's reciprocity law, which includes the quadratic reciprocity law as a special case, is the main result of class field theory, that is, the description of all abelian extensions of a given field F. In this talk we survey, in an elementary fashion, reciprocity laws from Euler to Artin. We finish with Langlands reinterpretation of Artin's reciprocity and how it leads to the formulation of a non abelian class field theory.

--

Reciprocity law,quadratic reciprocity,Langlands Program