We will identify the ring $\mathbb{Z}_{n}$ of integers modulo $n$, with the set $\{0,1,\ldots n-1\}$.

Lemma 1 extends easily from the Legendre symbol to the Jacobi symbol $\left(\frac{m}{n}\right)$ for odd $n$. The following is Zolotarev’s penetrating proof of the quadratic reciprocity law, using Lemma 1.

And finally, we proceed to prove quadratic reciprocity. Let $p$ and $q$ be distinct odd primes. Denote by $\pi$ the canonical ring isomorphism $\mathbb{Z}_{{pq}}\to\mathbb{Z}_{{p}}\times\mathbb{Z}_{{q}}$. Define two permutations $\alpha$ and $\beta$ of $\mathbb{Z}_{{p}}\times\mathbb{Z}_{{q}}$ by $\alpha(x,y)=(qx+y,y)$ and $\beta(x,y)=(x,x+py).$ Finally, define a map $\lambda:\mathbb{Z}_{{pq}}\to\mathbb{Z}_{{pq}}$ by $\lambda(x+qy)=px+y$ for $x\in\{0,1,\ldots q-1\}$ and $y\in\{0,1,\ldots p-1\}$. Evidently $\lambda$ is a permutation.

Note that we have $\pi(qx+y)=(qx+y,y)$ and $\pi(x+py)=(x,x+py)$, so therefore

Let us compare the signatures of the two sides. The permutation $m\mapsto qx+y$ is the composition of $m\mapsto qx$ and $m\mapsto m+y$. The latter has signature $1$, whence by Lemma 1,

and similarly

By Lemma 2,

Thus

which is the quadratic reciprocity law.

Reference

G. Zolotarev, Nouvelle démonstration de la loi de réciprocité de Legendre, Nouv. Ann. Math (2), 11 (1872), 354-362