Kth power residue chains of global fields

Abstract

In 1974, Vegh proved that if k is a prime and m a positive integer, there is an m term permutation chain of kth power residue for in- finitely many primes (E.Vegh, kth power residue chains, J. Number Theory 9 (1977), 179-181). In fact, his proof showed that 1,2,2 2 ,...,2 m 1 is an m term permutation chain of kth power residue for infinitely many primes. In this paper, we prove that for any possible m term sequence r1,r2,...,rm, there are infinitely many primes p making it an m term permutation chain of kth power residue modulo p, where k is an arbitrary positive integer. From our result, we see that Vegh's theorem holds for any positive integer k, not only for prime numbers. In fact, we prove our result in more generality where the integer ring Z is replaced by any S-integer ring of global fields (i.e., algebraic number fields or algebraic function fields over finite fields).