$k$th power residue chains of global fields

TL;DR

This paper generalizes Vegh's theorem on $k$th power residue chains, proving that any possible sequence can form such chains modulo infinitely many primes across various global fields, extending the result beyond prime $k$.

Contribution

It extends Vegh's theorem to all positive integers $k$ and to $S$-integer rings of global fields, broadening the scope of $k$th power residue chains.

Findings

Any sequence can be a $k$th power residue chain modulo infinitely many primes.

The result holds for all positive integers $k$, not just primes.

Applicable to $S$-integer rings of global fields, including number and function 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 $k$th power residue for infinitely many primes [E.Vegh, $k$th 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 $k$th power residue for infinitely many primes. In this paper, we prove that for any "possible" $m$ term sequence $r_1,r_2,...,r_m$, there are infinitely many primes $p$ making it an $m$ term permutation chain of $k$th power residue modulo $p$, where $k$ is an arbitrary positive integer [See Theorem 1.2]. 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).