We will identify the ring of integers modulo , with the set .
Lemma 1 (Zolotarev).
If is even, then
And if is odd, then divides , so
In both cases, the lemma follows from Euler’s criterion. ∎
Lemma 1 extends easily from the Legendre symbol to the Jacobi symbol for odd . The following is Zolotarev’s penetrating proof of the quadratic reciprocity law, using Lemma 1.
Let be the permutation of the set
which maps the th element of the sequence
to the th element of the sequence
for every from to . Then
and if and are both odd,
We will use the fact that the signature of a permutation of a finite totally ordered set is determined by the number of inversions of that permutation. The sequence defines on a total order in which the relation means
The only pairs that get inverted are, therefore, the ones with and . There are indeed such pairs, proving the first formula, and the second follows easily. ∎
And finally, we proceed to prove quadratic reciprocity. Let and be distinct odd primes. Denote by the canonical ring isomorphism . Define two permutations and of by and Finally, define a map by for and . Evidently is a permutation.
Note that we have and , so therefore
Let us compare the signatures of the two sides. The permutation is the composition of and . The latter has signature , whence by Lemma 1,
By Lemma 2,
which is the quadratic reciprocity law.
G. Zolotarev, Nouvelle démonstration de la loi de réciprocité de Legendre, Nouv. Ann. Math (2), 11 (1872), 354-362
|Date of creation||2013-03-22 13:28:25|
|Last modified on||2013-03-22 13:28:25|
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