Talk:Proofs of quadratic reciprocity

These things need to be done: Dmharvey Talk 03:10, 6 November 2005 (UTC)
 * 1) Clean up, wikify and check what I've written.
 * 2) Finish the bits of that proof that I haven't written yet (in particular the calculation of the quadratic subfield of the cyclotomic field via gauss sums, or whatever method).
 * 3) Add explanation of how to get the "supplementary theorems" in the same spirit as the proof I have partially written.
 * 4) Last but CERTAINLY NOT LEAST, write up a proof that doesn't use all that algebraic number theory!!! i.e. one that is accessible -- yes such proofs do exist!

Two elementary proofs you might like are Hammick 2001: http://www3.telus.net/ldh/math/qrl.html and Zolotarev 1872: http://planetmath.org/encyclopedia/ZolotarevsLemma.html

eisenstein's proof
I've added half of eisenstein's proof. The other half ("eisenstein's lemma") still needs to be written.
 * Yes Oleg, I know you think section stubs are ugly, but I'm tired now, and I promise I will make the stub notice disappear in the next few days :-)
 * done now Dmharvey 11:56, 14 April 2006 (UTC)

Dmharvey 02:42, 14 April 2006 (UTC)
 * I read about this proof at http://math.nmsu.edu/~history/eisenstein/eisenstein.html, but it would be nice to have a primary reference for it.
 * I don't know whether "Eisenstein's lemma" is a nonstandard name for this result, perhaps coined in the above-mentioned article, or whether it is is standard.

I think Eisenstein's proof is

Geometrischer Beweis des Fundamentaltheorems für die quadratischen Reste, J. Reine Angew. Math. 28 (1844), 246-248; Math. Werke I, 164-166

but I haven't checked. The textbook proof is a slight variation of E's original.

LDH 01:48, 22 April 2007 (UTC)

Would it be a good idea to add something on the 2nd supplement in the Eisenstein proof? It looks to me that the lemma and the lattice point counting idea work for this case too, but I don't have a reference for that and I'm not highly confident, so I haven't edited the page.

Gingercatnine (talk) 21:10, 28 January 2017 (UTC)

Gauss sums proof
I've changed references to Fermat's little theorem to the binomial theorem. Possibly Fermat was being used in a subtle way or an extension of it was being used, but it wasn't clear to me how this worked and it seems more likely that it referred to an idea from a proof of Fermat which is more clearly described by referencing the binomial theorem. I found the reference to Fermat made the argument harder to follow.

I've also added something about what it means to take algebraic integers mod p.

Gingercatnine (talk) 21:10, 28 January 2017 (UTC)