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 :
| 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 if and only if 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.
Read more about this topic: Zolotarev's Lemma
Famous quotes containing the word proof:
“Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“There is no better proof of a man’s being truly good than his desiring to be constantly under the observation of good men.”
—François, Duc De La Rochefoucauld (1613–1680)