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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.28 From Quadratic Reciprocity to Class Field Theory

 Kiran S. Kedlaya

The law of quadratic reciprocity, discovered by EULER [VI.19] and first proved by GAUSS [VI.26] (who dubbed it his theorema aureum, or golden theorem), is considered a crown jewel of number theory, and with good cause. Whereas its statement could be rediscovered by a sufficiently ingenious student (indeed, it actually has been rediscovered on a regular basis at the Arnold Ross mathematics summer program for several decades), rare is the student who comes up with a proof unassisted.

The law is most conveniently stated in a formulation due to LEGENDRE [VI.24]. For n an integer not divisible by the prime p, write () = 1 if n is congruent to some perfect square modulo p, and ...

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