# Quadratic reciprocity and Zolotarev’s Lemma

I want to explain a beautiful proof of the Law of Quadratic Reciprocity from c. 1874 due to Egor Ivanovich Zolotarev (1847-1878). Some time ago I reformulated Zolotarev’s argument (as presented here) in terms of dealing cards and I posted a little note about it on my web page. After reading my write-up (which was unfortunately opaque in a couple of spots), Jerry Shurman was inspired to rework the argument and he came up with this elegant formulation which I think may be a “proof from the book”.  The following exposition is my own take on Jerry’s argument.  The proof requires some basic facts about permutations, all of which are proved in this handout.

Let $m$ and $n$ be odd relatively prime positive integers.  You have a stack of $mn$ playing cards numbered 0 through $mn-1$ and you want to deal them onto the table in an $m \times n$ rectangular array.  Consider the following three ways of doing this:

Row deal ($\rho$) : Deal the cards into rows, going left to right and top to bottom.