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] History
This lemma was introduced by Yegor Ivanovich Zolotarev in an 1872 proof of quadratic reciprocity.
See also: Gauss lemma.
[edit] Reference
- E. Zolotarev, Nouvelle démonstration de la loi de réciprocité de Legendre, Nouv. Ann. Math (2), 11 (1872), 354-362
