Zolotarev's lemma
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In mathematics, Zolotarev's lemma in number theory states that the Legendre symbol
for an integer a modulo a prime number p, can be computed as
- ε(πa)
where ε denotes the signature of a permutation and πa the permutation of the residue classes mod p induced by modular multiplication by a, provided p does not divide a.
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[edit] Proof
In general, for any finite group G of order n, it is easy to determine the signature of the permutation πg made by left-multiplication by the element g of G. The permutation πg will be even, unless there are an odd number of orbits of even size. Assuming n even, therefore, the condition for πg to be an odd permutation, when g has order k, is that n/k should be odd, or that the subgroup <g> generated by g should have odd index.
On the other hand, the condition to be an quadratic non-residue is to be an odd power of a primitive root modulo p. The jth power of a primitive root, because the multiplicative group modulo p is a cyclic group, will by index calculus have index the hcf
- i = (j, p − 1).
The lemma therefore comes down to saying that i is odd when j is odd, which is true a fortiori, and j is odd when i is odd, which is true because p − 1 is even (unless p = 2 which is a trivial case).
[edit] Another proof
Zolotarev's lemma can be deduced easily from Gauss's lemma and vice versa. The example
,
i.e. the Legendre symbol (a/p) with a=3 and p=11, will illustrate how the proof goes. Start with the set {1,2,...,p-1} arranged as a matrix of two rows such that the sum of the two elements in any column is zero mod p, say:
| 1 | 2 | 3 | 4 | 5 |
| 10 | 9 | 8 | 7 | 6 |
Apply the permutation 
(mod p):
| 3 | 6 | 9 | 1 | 4 |
| 8 | 5 | 2 | 10 | 7 |
The columns still have the property that the sum of two elements in one column is zero mod p. Now apply a permutation V which swaps any pairs in which the upper member was originally a lower member:
| 3 | 5 | 2 | 1 | 4 |
| 8 | 6 | 9 | 10 | 7 |
Finally, apply a permutation W which gets back the original matrix:
| 1 | 2 | 3 | 4 | 5 |
| 10 | 9 | 8 | 7 | 6 |
We have W -1=VU. Zolotarev's lemma says (a/p)=1 iff the permutation U is even. Gauss's lemma says (a/p)=1 iff V is even. But W is even, so the two lemmas are equivalent for the given (but arbitrary) a and p.
[edit] History
This lemma was introduced by Yegor Ivanovich Zolotarev in an 1872 proof of quadratic reciprocity.
See also: Gauss's lemma.
[edit] References
- E. Zolotarev, Nouvelle démonstration de la loi de réciprocité de Legendre, Nouv. Ann. Math (2), 11 (1872), 354-362
[edit] External links
- PlanetMath article on Zolotarev's lemma; includes his proof of quadratic reciprocity
