整体域的$k$次幂剩余链
$k$th power residue chains of global fields
摘要：
1974年，Vegh证明了：若$k$是素数，$m$是正整数，则对于无穷多个素数，存在$k$次幂剩余的$m$项置换链[E.Vegh，$k$次幂剩余链，数论杂志，9（1977），179-181].事实上，他的证明表明，$1,2,2^2,...,2^{m-1}$是一个$k$项置换链的$k$次幂剩余的无限多个素数。本文证明了对于任何“可能的”$m$项序列$r_1,r_2,...,r_m$，存在无穷多个素数$p$使其成为$k$次幂剩余模$p$的$m$项置换链，其中$k$是任意正整数[见定理1.2].从我们的结果中，我们看到Vegh定理对任何正整数$k$都成立，而不仅仅是对素数。事实上，我们证明了我们的结果在更一般的整数环$\Z$被替换为任何$S$ -整数环的整体域（即代数数域或代数函数域有限域）。
英文摘要：
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).
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