Editing Quadratic reciprocity (section)

==Connection with cyclotomic fields==

The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used [[Gauss sum]]s to show that [[quadratic field]]s are subfields of [[cyclotomic field]]s, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomic fields. His proof was cast in modern form by later algebraic number theorists. This proof served as a template for [[class field theory]], which can be viewed as a vast generalization of quadratic reciprocity.

[[Robert Langlands]] formulated the [[Langlands program]], which gives a conjectural vast generalization of class field theory. He wrote:<ref>{{cite web|url=http://www.math.duke.edu/langlands/Three.pdf |title=Archived copy |access-date=June 27, 2013 |url-status=dead |archive-url= https://web.archive.org/web/20120122104607/http://www.math.duke.edu/langlands/Three.pdf |archive-date=January 22, 2012 }}</ref>

:''I confess that, as a student unaware of the history of the subject and unaware of the connection with cyclotomy, I did not find the law or its so-called elementary proofs appealing. I suppose, although I would not have (and could not have) expressed myself in this way that I saw it as little more than a mathematical curiosity, fit more for amateurs than for the attention of the serious mathematician that I then hoped to become. It was only in Hermann Weyl's book on the algebraic theory of numbers<ref>{{Cite book |isbn = 0691059179|title = Algebraic Theory of Numbers|last1 = Weyl|first1 = Hermann|year = 1998| publisher=Princeton University Press }}</ref> that I appreciated it as anything more.''