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What is a conceptual proof of quadratic reciprocity? I saw the wiki proof using algebraic machinery. I don't know what field extensions, Galois groups etc are. But I want to understand how properties under different moduli primes are getting related. I always wanted to understand how can one use all the modulo p information to understand globally. I would like to know if someone can explain quadratic reciprocity in a way that I will understand.

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    Why not learn about field extensions, Galois groups etc. in order to understand the proofs you tried to read? That machinery is important for many other things besides a proof of quadratic reciprocity. It will be time well-spent. In the meantime, indicate what your background is (not just what it is not) so that people who want to indicate other proofs of QR can know what math belongs in a proof that you currently will understand.2011-10-30
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    See http://mathoverflow.net/questions/1420/whats-the-best-proof-of-quadratic-reciprocity. Rousseau's proof is quite conceptual.2011-10-30
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    Gauss's first proof (as it appears in the *Disquisitiones*) is *very* "hand on", and you can really see the recursion working. It relies on a couple lemmas about being able to find certain residues and non-residues among primes, and it is rather lengthy (breaking into several cases, subcases, and sub-subcases), though.2011-10-30

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