Let $f\in \Z[x]$ be an irreducible monic polynomial of degree n > 0 with integer coefficients. Given a prime p, reducing the coefficients of f modulo p, gives a new polynomial which can be reducible. A reciprocity law is the law governing the primes modulo which f factors completely. The celebrated quadratic reciprocity law, introduced by Legendre and completely solved by Gauss, is the case when f has degree two. Many other reciprocity laws due to Eisenstein, Kummer, Hilbert and others lead to the general Artin's reciprocity law and (abelian) class field theory in the early 20th century. In 1967, in a letter to André Weil, Robert Langlands paved the way for what is known today as the Langlands Program: a set of far reaching conjectures, connecting number theory, representation theory (harmonic analysis) and algebraic geometry. It contains all the abelian class field theory as a particular case, and another special case plays a crucial role in Wile's proof of Fermat's Last Theorem. There is a vast amount of number theory problems than can be studied in the framework of the Langlands Program, namely: (i) non-abelian class field theory; (ii) several conjectures regarding zeta-functions and L-functions; (iii) and an arithmetic parametrization of smooth irreducible representations of reductive groups. In this talk we will give an elementary introduction to the Langlands Program, dedicating special attention to the local Langlands correspondence and explain how it can be seen as a general non-abelian class field theory. We shall concentrate more on examples, avoiding general and long definitions. If time permits, an application to noncommutative geometry will also be presented.

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Class field theory,Langlands correspondence,L-function