A remark on Zoloterav's theorem
Abstract
Let n>=3 be an odd integer. For any integer a prime to n, define the permutation gammaa,n of 1,...,(n-1)/2 by gammaa,n(x)=n-axn if axn>=(n+1)/2, and axn if axn<=(n-1)/2, where xn denotes the least nonnegative residue of x modulo n. In this note, we show that the sign of gammaa,n coincides with the Jacobi symbol (a/n) if n=1 mod 4, and 1 if n=3 mod 4.
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