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Primes, permutations and primitive roots. (English) Zbl 1230.11005

To a primitive root \(g\) modulo \(p\) we associate the permutation \(\sigma_g\) of \(X:=\{1,2,\ldots,p-1\}\) defined by \(\sigma_g(x)\equiv g^x\pmod p\). More precisely, \(\sigma_g(x)=y\), the unique element in \(X\) satisfying \(y\equiv g^x\pmod p\). For example, if \(p=7\), then \(\sigma_5=(1~5~3~6)(2~4)\). The authors determine the sign \(s(\sigma_g)\) of the permutation \(\sigma_g\). Assume \(p>3\) and put \(w=((p-1)/2)!\). Then, modulo \(p\), \(s(\sigma_g)=-wg^{(p-1)/4}\) if \(p\equiv 1\pmod 4\) and \(-w\) otherwise.

MSC:

11A07 Congruences; primitive roots; residue systems
11R29 Class numbers, class groups, discriminants
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